Do you want "constant" as in "independent of output voltage", or "constant" in absolute terms?
The first is easy, a current mirror will do the job. The output current will be fairly independent of output voltage (give or take Early effect, which still won't be too bad at that voltage, and as long as it stays out of saturation, say Vce > 0.2V), but depends directly on the input current. If you use a resistor to supply said current, then it's linearly dependent on supply voltage.
Tim