Cross sections of 1n-removal reactions from the neutron-rich nucleus (37)Mg on C and Pb targets and the parallel momentum distributions of the (37)Mg residues from the C target have been measured at 240 MeV/nucleon. A combined analysis of these distinct nuclear- and Coulomb-dominated reaction data shows that the (37)Mg ground state has a small 1n separation energy of 0.22(-0.09)(+0.12) MeV and an appreciable p-wave neutron single-particle strength. These results confirm that (37)Mg lies near the edge of the "island of inversion" and has a sizable p-wave neutron halo component, the heaviest such system identified to date.