Question

what path will moon take when the stops rotating round the Earth?

Answer 1

When the moon will stop revolving around the earth (if that's what you meant), then it will have an orbit of its own and will start revolving around the sun like the other planets do. 

Answer 2

If the moon stops revolving around the earth it will leave the orbit around earth and will make its own orbit in solar system of sun and will revolve around the sun.
It means like other planets e.g. earth and mars etc which are revolving around the sun as moon is also a planet so it will also revolve around the sun.