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1 mA is not that low... with 1kOhm load you are already at 1 V which would be a quite high voltage for a microcontroller.
Dependent on your current source (a sensor?) and how much resistance it can drive you could in the simplest scenario just terminate with a resistor.
What's the lowest value that you have to convert? You might have to consider the noise contribution from the amplifier/resistor.
You could also use a more dedicated CSA (current sense amplifier) or simply drive the current into the negative node of an opamp with a feedback resistor across output and negative input. A voltage on the positive input would be used to fixate the DC level at the output of your current source. The voltage across the feedback resistor, essentially the opamp output is then your amplified voltage:
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