There are six distinct solutions to this puzzle. Rotations and reflections of these will also be valid. The integers can be around the circle in the following orders:

1 6 11 2 15 8 3 16 13 18 5 12 17 14 9 4 7 10
1 6 11 2 15 8 3 10 13 18 5 12 17 14 9 4 7 16

1 10 3 8 15 2 11 6 13 16 7 4 9 14 17 12 5 18
1 16 3 8 15 2 11 6 13 10 7 4 9 14 17 12 5 18

1 12 5 2 15 8 3 16 13 18 11 6 17 14 9 4 7 10
1 12 5 2 15 8 3 10 13 18 11 6 17 14 9 4 7 16