Tony is right on the money with this. That's what I was trying to get at a few posts ago when I was saying if the tempo was off just a fraction of a BPM it would sound in sync initially but would get more and more out of sync as the song progresses.
I had to go through the exact same process a while back when my songwriting partner sent me a piano track that was slightly off BPM wise; I set reaper to the BPM of her file, loaded it in, then set the project tempo to the proper BPM, reaper does the rest.