Before revision:  We prove that if {\cal  {\cal G} is minor-closed, then there is{\cal  {\cal  {\cal  {\cal  {\cal   an algorithm that (k)}\cdot n^3-time and } either returns a set S certifying that G is a k-apex of {\cal  {\cal G}  or reports that such a set does not exist, in 2^{{\sf poly(k)}\cdot n^3 time} \notin {\cal  {\cal   .
After revision:  We prove that if {\cal {\cal G} is minor-closed, then there is{\cal . In the first paper of this series we obtained upper bounds on the size of the graphs in the minor-obstruction set of{\cal {\cal {\cal {\cal an algorithm that (k)}\cdot n^3-time and } either returns a set S certifying that G is a k-apex of {\cal {\cal G} or reports that such a set does not exist, in 2^{{\sf poly(k)}\cdot n^3 time} \notin {\cal {\cal .