starter motor problems?

When my 89 LX is cold, the car starts no problem. But, when it warms up and i stop the engine, then try to start it again, the starter motor struggles to turn over the engine. It has the same symptoms of a dead battery, where the starter motor is struggling to crank the enigine.

However, if i let the car cool down, it starts perfectly. I have replaced the battery and the problem continues. Does this sound like a dying starter motor?

These forums are great, thanks for any advice you might have on this problem.