We have estimated the potential curves of the Mg(3s(2)) . Ne((1)Sigma(+)), Mg(3s3p) . Ne((3)Pi, (3)Sigma(+)), Mg(3p(2)) . Ne((3)Sigma(-)), Mg+ (3s) . Ne((2)Sigma(+)), Mg+(3p) . Ne((2)Pi), and Mg+2(2p(6)) . Ne((1)Sigma(+)) van der Waals states by means of ab initio calculations. Similar to the analogous doubly-excited states of MgAr and MgKr, the Mg(3p pi 3p pi) . Ne((3)Sigma(-)) state is found to be unusually strongly bound, D-e = 548 cm(-1), a bond strength which is more than 20 times that of the singly-excited Mg(3s3p pi) . Ne((3)Pi) state and even more than three times that of the Mg+(3s) . Ne ion. The strong bonding is attributed primarily to the lack of a Mg( 3s) electron, so that all the attractive van der Waals forces can extend to smaller internuclear distances because there is no Mg(3s sigma)/Ne(2p sigma) exchange repulsion.