If you are using a digital multimeter to check the hot to ground voltage, and are seeing 60 volts, then chances are the ground wire is totally disconnected somewhere between the breaker panel's ground buss strip and the place where you're testing. Tthe voltage reading you are getting is probably just occurring through capacitive coupling between the ground wire and the neutral wire. You definitely should get 120V between black and ground if the ground is as it should be.
Since you are getting as much as 60 volts, I'd bet that you won't find the break right at the outlet you are checking, I'd think there would have to be quite a few feet of wire to have enough capacitance to push the meter that high on a 60 Hz circuit.
Look for and fix the break in the ground lead.
Let us know what you find.
Happy New Year,
Jeff