Unless I'm mistaken in reading the LT1013 datasheet, in single supply operation it can swing within a couple mV of ground.
The LT1013/LT1014 are fully specified for single supply
operation, i.e., when the negative supply is 0V. Input
common mode range includes ground; the output swings
within a few millivolts of ground.
I was looking at the output voltage swing specified for R
L=2k and +/-15V supply which is not the case here, oops...
Anyway, xReM1x, here's a version that will work. You had R1 set too low, giving you 0.1V/1A at the output. You could go with an op amp for current sensing but your accuracy will depend on the op amp's input offset voltage and the matching of the four resistors of the difference amplifier. LTC6101 is the easy way here and it will work.
EDIT: V3 should be 0.5V not 5V.