If you want to vary the speed and still keep it accurate, you need some form of motion controller. Google that and you'll get bazillion hits for all sorts of motors. It will include encoder or taco feedback, and you may need to change the motor. Look around the robot supply shops (battle bots and the like). I imagine you could find something for around $200.
If you want to have 6rpm now and for ever, well at least for a while, you have two options for regular speed control.
1) DC Motor running at it's rated speed, from a regulated supply, with a gearbox to give you the speed reduction. The motor speed and gearbox size will depend on how big a gearbox you can fit/afford. The idea here is to add inertia to the system (flywheels work well to). Get the motor pumping with a large reduction and any small changes in supply voltage won't have a big effect on your work piece.
Even adding a larger flywheel/pulley to the lathe chuck would help. The faster you can run the motor the better.
2) Use an AC Motor, again with a gear box to reduce the speed (a 200:1 reduction on a 6 pole motor I believe). This will give you better speed regulation because the speed is based on the mains frequency, not the voltage, but speed changes require a different hardware reduction.
Both these methods will protect your workpiece when the TIG fires up as well.
Good luck