It's probably your ballast...a component of the light fixture that the power goes through before getting to the bulbs.
If you're "lucky", it will be a small screw in ballast like shown here
that cost about $6...although I think these might only be used in single bulb fixtures. If you don't see one like this, then it's probably one like this
<- click). That site has instructions on the labor portion of replacing a ballast. To buy the actual ballast, take it to the lighting department of your local Home Depot or Lowe's...they'll help you pick the correct one. Last I looked, they were around $35 or so for a less efficient magnetic ballast, $40 or so for the more efficient electronic ballast.
...of course...this assumes that it is the ballast...maybe someone could verify that if one bulb goes out, it doesn't cause both to go black.