Current sense amplifier + op-amp buffer + ADC: Measuring down to 0 with a single supply
You can get a small negative voltage by using an LM7705 which produces -232mV nominal output voltage using a charge pump. 



The advantage of using that part over a typical garden-variety inverting charge pump converter (eg. +5 to -5) or inverting boost converter is that the worst-case negative output voltage generally falls within the maximum negative input voltage of something like your ADC converter (-300mV in your case, which is typical), so you don't need to try to clamp the op-amp output/ADC input voltage near ground. 

On the other hand, it's probably more expensive than some other solutions that would take more engineering effort, so this is just one of many possibilities.