Hi,
Assuming you can allow a small resistance I would suggest you consider one of these 24bit A/D with a suitable low value resistor.
If I read the spec correctly at a gain of 128 the input range is +/-9mv . If you used a 0.1 ohm sense resistor you would have ~ 0.4mv full scale. Which means you only get to use 18 bits of the 24 bits. Still pretty good.
For the resistor, I've used something like this Resistor in the past for similar measurements.
Note: with any precision you are seldom able to attain the "calculated" results but I think this will work for you. It would be even better if you can increase the 0.1 resistor value.
If you need absolute values, calibration is another issue.