A Motor That Only Runs One Rotation (Or For A Certain Amount Of Time)
If you need to reliably turn once, then a time function is not going to work - as conditions change (battery voltage, temperature, the slow buildup of gunkies in the mechanical components), the amount of time for "one rotation" will be greater or less than it was the day you built it.  

You want to detect "the motor has completed its path" ... There are several techniques.  One popular, cheap, and easy one: There are "photo-interrupter" modules (basically a block of epoxy with a small IR LED pointed at a small photo-diode or photo-transistor) that you can pass a notched disc through.  When the notch is in front of the photo-diode, you get a signal that would tell your circuit "stop now, please".  

A simple R-S flip-flop (built out of two sections of a NAND or NOR gate) and a transistor to control the motor current would complete the circuit.  (one minor gotcha -- the 'begin turning' pulse must be shorter than the time it takes to make a rotation ... if you can't guarantee that, a second chip can solve that problem)

Sparkfun has a "kitlet" with a photo-interrupter ... https://www.sparkfun.com/products/9322