Hi,
Are you saying that the current is between 0 and 1mA? If so I'd say that explains it since the sensitivity of the sensor, for the X05B version of the ACS712, is 185mV/A nominally which for 1mA means an change in output voltage of 185uV.

If you're saing that the current is from 0 to 1A then you should see the output voltage of the sensor change from the 2.5V nominal to 2.685V OR 2.315V depending on the direction of current flow.

If you have the X20 or X30 version the sensitivity is even lower at 100 and 66mV/A repectively.

/Henrik.

EDIT:Alain beat me to it....