LM317 to create constant current of 2mA?

I have a circuit where I want to have a constant current of 2mA through a variable resistor. I've been told that I could probably use a LM317 as a current regulator, with one resistor on the ouput. But I've read some places that the LM317 takes minimum 5-10mA load to function correctly.

How can I achieve a constant current output of 2mA when I don't know the resistance of the variable resistor?

The input voltage is about 2.755V. Output voltage doesn't matter, just the current.

Here's an image to my feeble attempt at a schematic:

The LM317 with the single series resistor between output and adjust input is actually a fixed current source, not a current limiter. You don't need the LM317 to create a current limiter, a few discrete components will do:

For a limiting at 2mA you select a 330Ω resistor for RSENSE. If there flows 2mA through it Q2 will start to conduct and reduce the base voltage of Q1, so that its current is cut off.

edit (re changed question)

Maybe you're focusing too much on the LM317. If you need a constant current you could use the LM234 which is a programmable current source for up to 10mA. You set the current with a resistor.