Sorry to join the party late. This does sound like a bad starter but here are a few things to consider before replacing it.

If it is a relatively new fixture it may not have a separate starter and the best fix would be replacing the entire fixture. If it does have a removable starter (small metallic cylinder usually) you can replace it pretty easily - just get a replacement unit with the correct wattage rating. The starter is a pretty simple device and usually does not contain any toxic chemicals (unlike the ballast or the tube). Check with your local waste disposal folks but around here they just say to toss them.

Since you have replaced the tube and it is still acting up it is very likely that the ballast has been damaged also. I don't remember the details on why but trying to run a fluorescent fixture with a failed (failing) starter often kills the ballast too. Ballasts are nasty beasts and must be disposed of properly. Replacing one usually involves taking the fixture apart, cutting the old one out and wiring the new one in with wire nuts.

If you go that route please be careful.

-Eric