I have an audio file of a song that I'd like to learn and have dragged into Studio One and at some point may want to mix BiaB content with it. I don't know the exact tempo of it but suspect it's somewhere around 83 BPM. S1 has a time stretching capability that I have used in the past and therefore believed that I understood it; but apparently not. Here is the procedure that I have successfully used in the past.

1. Drag in the audio
2. Display “Bars” in the Time Ruler
3. Drag the beginning (first down beat) of the song to Bar #2
4. Activate the Metronome
5. Play the audio and adjust the tempo to match
6. When happy with the tempo, right-click the audio and double-click value in the File Tempo field (it may say “Not Set”). Type in the tempo value. The Speedup field should be set at 1.0
7. To change the tempo of the audio adjust the Speedup value.

My problem starts at Step 5. I can get the tempo of the song and the tempo of the metronome to match only for the first few bars, then they drift apart.

Note in the screenshot that bar 2 of the waveform is aligned with bar 2 of the ruler and the same for bar 3. Bar 1 is reserved for the 4 beat count-in.
This is from the manual.
Automatic timestretching is based on the relationship between the Song tempo and the audio file’s tempo.
Each Audio Track has a Tempo mode that controls the behavior of the Events on the Track, based on the Song tempo. The Tempo mode can be selected in the Track Inspector. The following modes are available:
•Don’t Follow Events on this Track are independent of the Song tempo. They are never moved or stretched automatically.
•Follow The start positions of Events on this Track are tied to the musical grid. Thus, the Events move when the Song tempo changes but they are not stretched.
•Timestretch Event start positions follow the Song tempo, as in Follow mode. In addition, the Events are stretched to fit the Song tempo.

Any idea what I'm not understanding here?

If the source files has a varying tempo then you'll need to tempo map it, if it's a constant tempo then you can use the Alt key and drag the end of the file to align with the beats.

Thanks for informing me about the Tempo Map; never heard of it before.
So I displayed it and it does show it at the top of the screen but it doesn't seem to be analyzing my audio waveform to determine the tempo(s). It displays one tempo value, which after some exploration, is the same value set at the bottom of the screen.
Is there a way for it to determine what the tempo(s) are throughout the song?
If yes, can I then specify a constant tempo which it would then alter my audio to play at?

See video, if you have Melodyne with Studio One you can get the tempo map, if not you can use the Audio Chord Wizard in RealBand to map out the tempo, then create a midi track in RealBand and open it with Studio One, then drop the source file in and it will align with the beats. You can then drop acidized files from Biab that will fit to the tempo map or you can change the map to a constant tempo.

Many thanks for the help.
This video is EXACTLY what I'm trying to do; take a free-tempo audio file, determine it's tempo map and then convert it to a static tempo so I can work with it., and I do have Melodyne installed (see screen shot].

Captain Gregor [in his starship ] and I diverge on 2 points.
1. I get no "Monophonic Detection" progress clock happening; 2:40 in the video.
2. "Map" doesn't show up in the "File Tempo" field.

Any idea on why we diverge?

If you can't get it in melodyne try the RealBand ACW method.

Success!!
A ton of thanks Musocity. I shut everything down, started from scratch and followed "Captain Gregor's" instructions and it worked like a charm.
A common occurrence (for me) in doing a new set of tasks with unfamiliar software (Melodyne) is not knowing what to expect at each step, clearly I bungled something up.

What these 2 tools did was analyze an arbitrary audio waveform to determine what the tempo in BPM is at each point in time; to no less than 4 decimal places! This "tempo map" can vary wildly throughout the song especially if the band is playing live. [I happen to know something about this from experience when playing in a church band and listening to a recording of us. Almost without exception, we would all speed up during the high-energy chorus and then slow down when the softer parts came. Playing to a clicktrack is not easy and our drummers couldn't do it.]

Back to S1: With the software having a knowledge of what the map is, the user can then alter it at will and the audio will then follow your new map. For me, I just want a static BPM for the whole song.

I don't know what you think, but I'm blown away by the sophistication of the algorithms that have been programmed to accomplish this. Very impressive from a computer science perspective.
I wonder if at least a high-level description of how they pulled this off is available to the public. I don't think AI is involved in any way and I'm guessing that somewhere there is a public domain paper written by a grad student or musicologist/researcher on the subject.

Thanks again for the videos . . . this is good stuff!

Nearly all current DAWs have this ability nowadays, with or without Melodyne but Melodyne can make it easier. For those DAWs without Melodyne Essential, there are ways to obtain it inexpensively bundled with other plug-ins from iZotope and others. There's generally no reason to pay $99 just to get it directly from Celemony.

Dirty little (not so) secret—ok, worst kept secret in the music industry: iZotope has no way of verifying old licenses if you buy upgrades through a reseller. $99 for Nectar 4 Standard Upgrade or $119 for Nectar 4 Advanced Upgrade not only gets you the app but includes a license for Melodyne Essential that you redeem at Celemony. Even better is that, should you want to upgrade to a higher version of Melodyne, you now have that upgrade path, too.

Now that iZotope is part of Native Instruments, I expect this upgrade loophole to close for good but, as of now, it's still open (as it has been for 20 years).

Audacity has now added tempo maps and time stretching.

Using Melodyne is one way of making a tempo map, and great when it works, but doesn't always give you the best results if the audio is very free and not enough transients to mark the beats, or there are lead instruments that are loud that are ahead or behind the beat. If Melodyne doesn't make a good map, it is easy in Studio One to just open the tempo track at the top, and slide the bars (or beats) to the position in the audio track that correspond to that bar or beat. You have to hold Ctrl while you slide the beat position to the left or right. The easiest way is to have the metronome on, and play the track and hear when the metronome gets out. Then take your mouse to the beat or bar in the timeline, and position it just below (inside the tempo track) and hold control and slide it to the right position (using your eyes on the audio track, or just listening.) After that, as you say, you can make a BIAB steady BPM follow your free tempo track.

PG Product's Realband program comes with a feature called Audio Chord Wizard .. which sounds like it is to analyze chords (which it does) but it is even better at adjusting a Tempo Map.
It comes free with your BiaB purchase.

It is much simpler in that you can either:
Play the song and tap the F8 key on the one of each measure to adjust, or
Drag the bar line to align where you want it.

Both of these options are available after Realband has already ran through your song and adjusted the Tempo Map to what it sees .. and it usually gets pretty darn close on its own.
And it gets pretty granular .. down to tempos like 121.344356.

"Captain Gregor's" solution worked very well for me.
I've detailed it below.