Distance sensor needed: 1mm resolution at 3cm - 6cm range. Any suggestions?

Hi guys!

I'm working on a project that requires me to periodically (on the order of once per second) measure the distance from the sensor to a fabric surface 3 to 6 cm away, with roughly mm accuracy. This will be mounted indoors, with fairly constant light levels, and no extreme environmental conditions. It would be great to have the sensor hooked to the Arduino, but the project is using a dedicated PC in communication with the Arduino, so if it had to run through the PC, i could live with it.

Does anyone have any suggestions before I dive in and start searching?