Ive built a current sense differential amplifier that measures the voltage across a 1 Ohm resistor to give me a reading for the current through it.

Ive connected it as follows, all resistors are matched 10K to give a gain of 1:



I am powering it from a single rail 12 volt source. The problem is even though this is a rail to rail opamp (Output Low swing from 5-20mv) im getting a constant error of 0.62v and it seems the opamp is not able to go below this value. 

I have tired to resolve this by connected Vout to ground using a 10K resistor but that makes no difference what so ever.

I tried it with dual supply configuration (+8v,-8v) and that seems to resolve the issue. But i want to run this off a single supply rail so this is not a viable option.

How can i resolve this? Im trying to measure the voltage drop across a 1ohm resistor to give me a reading for current through the circuit (0-1v represents 0-1 Amp) so i can set up a current limit. A better way (or practice) to doing this is also welcomed.The initial (but not ideal) fix i can think of is to offset the readings by a few volts?

Edit:

Changed R1-R4 from 10K to 100K and im down to about 30-50mv! 


Question: What are the advantages and disadvantages of thinner PCB thickness (<1.6 mm or 0.063'')?
Answer: And the obvious one : smaller end product! If you're making a digital watch, 1.6mm is huge! MP3 players, wearable electronics, possibly cameras, phones etc similar. At these board sizes, flimsiness is not a problem.