Edge 305 shuts off during a short 1 hour ride

I have the same problem. When I contacted Garmin they quickly diagnosed that my battery disconnects as a result of vibrations (like the ones you feel on your handlebars). Since I purchased this device over a year ago, their solution was for me to send it to them and pay $100 to get it fixed. Now one of the guys I ride with is having the same problem. I am very disatisfied with the Garmin 305's reliability. I'm not giving Garmin any more cash. (I paid $350 for this GPS 18 months ago!)

Does anyone know how to open the 305 so I can resecure the battery?

I have had the same problem. I could tap it a few times and it would go off. Also, if I applied a little pressure to the sides, near the top of the GPS it would shut off. I just finished opening it and putting it back together.
I carefully removed the rubber using a small flat head screwdriver to get under it. I started at the bottom and pulled it away slowly with my fingers. The two parts of the GPS case are glued together and come apart fairly easily. Inside are little pins making contact with the battery side. I bent them up a little and used silicone to stick it back together. I am waiting for the silicone to dry and I'm trying it out.