This isn't necessarily a bear. I used a frog but music bear sounded much better.
This isn't that hard to make and all you need is a 9v batter, voice module (both from Radio Shack), box cutter or scissors, string to sew with, a needle, a speaker, and a stuffed animal.
Step 1: Cutting Open Your Stuffed Toy
Make sure you pick something that you know won't be a problem. I had trouble trying to get it into a stuffed cat and ended up having to use a frog that wasn't as stuffed.
You can use a box cutter or scissors. I used scissors since it was easier. Cut somewhere where the 9v battery can fit easily.
Step 2: Recording Your Song
My voice module doesn't have great audio quality, but I'm sure most of them don't. Connect the battery to the module. Pick any song of your choice and record it while holding down the white button (or whatever button is by itself). You can't hold it too close to the speaker or else it won't sound any better, so I recommend holding it a couple of inches away from it.
Step 3: Sewing
Once you get your song, put your voice module in your stuffed animal and be careful. Don't place the recording button anywhere near where a kid might accidentally press it again and it won't have a song.
This part isn't hard, you just have to sew the hole in your animal (ouch) closed.
Step 4: You're a Build-A-Bear Now!