Downscale a 0 - 5volt supply to a millivolt range
Just use a voltage divider, assuming a bit of source resistance (50-100 ohms, say) is acceptable. 

If you divide to .365/5 then you will still only use a portion of the DAC range, but it will be simpler. (easiest method)

If you divide to 145mV for 5V change at the input and add 220mV (three resistors and a reference voltage) you can use the entire range of the DAC.

Say you use a 49.9 ohm resistor to GND and you have a 5V reference available. Then you could use a 1034.3 ohm and 1635.9 ohm resistor to 5V and DAC output respectively. If I did the sums right that would give you 0.220 to 0.365V output for a DAC output 0~5V.   

If you need to  have an accurate output with mA of current flowing (as on the later comment) an op-amp buffer could be used. You can still use the voltage divider as above but it would be better to multiply each value by something like 10. Then a low offset op-amp voltage follower with positive and negative supplies. 1 LSB represents 141uV do pick an op-amp with maybe 50 to 100uV maximum Vos.  There will be many such suitable op-amps that can operate from +/-5V supplies.