ONE amp? That's not much .... especially considering the resolution of the meter. You're in a position where you're trying to measure ounces with a gallon jug.

More to the point, there may not be a problem at all. Electricity takes ALL paths, so some current will travel through the ground rod, the water piping, and every other ground, as well as taking the PoCo neutral back to the transformer.

To trace down the problem requires more time, and expertise, than even the ordinary sparky. It could be a utility problem, a problem with a neighbors' house, etc ..... or no problem at all. Though it might be a clue that you inspect the service drop for squirrels nibbling on the bare neutral.