I have had a GP-1 for a couple of years and find it absolutely invaluable. When traveling, I like to be able to pinpoint where I took my pictures and this enables me to do so without taking copious notes, etc.
I did not realize till recently that there was a GP-1A, which B&H advertises is specifically for the D600. I notice that this costs somewhat more than the GP-1. And the GP-1 is still available.
I initially used this GPS device with my D90, then the D7000. When I got a D600 I used it with it as well. In fact, the D600 manual specifically says the GP-1 is supported (page 175). I was surprised to see the GP-1A and am not sure what the difference between the two devices might be.
All I can say is that the GP-1 works with the D600. I use it all the time and would not be without it. The device is incredible easy to use. You just connect it. It turns on when the camera turns on. It may take a minute or two to find satellites, but I've never found it to be an issue. The initial locate takes the longest.
Some may complain that it drains the battery faster. Maybe so - I have never found it to be an issue. Any cons, in my opinion, are more than outweighed by the pluses.