Measuring current, instrumentation or differential amplifier?
For this specific project, which is current sense, the best option would be to select a Current Sense Amplifier. Those amplifiers are designed for exactly the desired applications. 

For example take a look at LTC6102, page 22 shows an example to drive a 16bit ADC. Calculations can be changed to support 100mOhms resistor.

If you are using LM4040 as ADC reference, make sure you get the "A-Grade" part with 0.1% tolerance which translates to 0.5mV. The parts with more than 0.2% Tolerance will already set your accurarcy 1mV off.