That's a very weak signal. What kind of sensor is it?
How much precision do you need?
You can read down to about 1mV with 1mV of resolution if you use the optional 1.1V reference. You can use a lower external reference but I don't know how far you can "push it".
The "perfect solution" would be an op-amp with positive & negative supplies. There are "rail-to-rail" op-amps but I'm not sure how well they work when you get THAT close to the rail (close to 0V) with a single-ended power supply).
Or, you could bias the op-amp. For example, with 1V of bias 5mV becomes 1.005V, etc. (not including amplification). (I don't have a schematic handy for you).
...This is usually tricky because with a weak signal you're usually starting with a poor signal-to-noise ratio. The noise gets amplified along with the signal, the amplifier generates some noise of it's own, and it may pick-up additional noise through the power supply or from electromagnetic interference, etc.