what you want is an op amp with a common mode input and output range including the negative rail (generally listed in the fist page of schematics)
as your signal level is 0-1V its very easy, just make sure your input range is better than vcc-2V, generally this means only 1.5V Maximum on the really cheapy ones so it shouldnt be an issue, for the input,
however for the output level, you run into an issue, if your aiming to make the circuit as simple as possible with a 1 Ohm measuring resistor so the signal levels match, then a vcc-1.5 wont cut it, and you will have to spend a few more cents, as at 1A your resistor is at 1V, when you add on the base-emmittor junction of the resistor this comes to 1.75V which is higher than the cheapy can handle,
short version: an op amp with an input range of 0-greater than 1, and an output range of 0 to greater than 2, those are the specs you want to make it work,