You're in luck in that this is a common issue with 240's so there's alot of good info out there. (I had to fix mine about a year ago after I pushed in the trip counter once while in motion - big mistake-and the odometer stopped right on the spot.) There's a tiny gear in there which becomes brittle with age and doesn't take much for just one of those little gear teeth to break off and then your odometer will come to quick halt.

Here's a couple of good links you can take a look at. This first one is the resource I went by when I replaced mine. Took maybe an hour or two and not hard at all if you just take your time, mark all the wires on the back to keep track and don't loose any little screws along the way! I bought my replacement gears from this site as well:

http://www.odometergears.com/documentation/Volvo_240_Odometer_Gear_Replacement.pdf

I decided to do both gears while I was in there. Figured it was worth a little bit more $ to do a permanent fix rather than have the bigger gear break on me sometime down the road.

Also for further info:
http://www.davebarton.com/240-odometer-repair.html

fcpgroton and IPD sell replacement gears as well:

http://www.fcpgroton.com/product-exec/product_id/17793/nm/1986_1993_Volvo_240_Odometer_Gear/category_id/161

http://www.ipdusa.com/products/6086/109758-240-odometer-gear-no-instructions