The odometer is driven by a small plastic gear which eventually breaks, causing the whole thing (trip and all) to stop moving. It's an easy repair if you're so inclined and you will save BIG $$$ b/c you won't need to buy a new odometer for $200+ at the dealer. Here's what to do:

You need to remove the entire instrument cluster (volvospeed.com has the instructions)

Take apart the cluster nice and easy after removing it. Eventually you'll get to the odometer. Now take apart the odometer (it's not that complex, really) and you'll see the broken gear. If you can find a replacement at a local speedo shop, you'll be in luck. I paid $40 (!) for that little gear at a shop over here in Southern California. Put everything back together and you should be back in business. Expect to spend an afternoon for a first time job. It would also be a good time to replace the bulbs, b/c mine burnt in less than a week after replacing the odometer gear. (I think old bulbs are vulnerable to movement and shock). Good luck!