current is more than enough, the uno by itself is pretty much nothing (until you go to run if off a battery, then it seems large), the regulator on board which is converting input to 5 volts is limited to (I think 500ma)
you will notice the regulator getting warm if use you 12 volts, cause (12-5) there's 7 volts that has to go somewhere, but that is acceptable... general thinking says if you can hold your finger on it for 5 seconds or more its fine, though it might alarm you if your thinking stuff should not be toasty (it shouldn't but there's math behind how toasty is too toasty, too toasty is darn near baking cookies in this case)