  A globally correct potential energy surface (PES) for the \hp\ molecular ion
is presented. The Born-Oppenheimer (BO) \ai\ grid points of Pavanello et. al.
[\textit{J. Chem. Phys.} {\bf 136}, 184303 (2012)] are refitted as BOPES75K,
which reproduces the energies below dissociation with a root mean square
deviation of 0.05~\cm; points between dissociation and 75\,000 \cm\ are
reproduced with the average accuracy of a few wavenumbers. The new PES75K+
potential combines BOPES75K with adiabatic, relativistic and quantum
electrodynamics (QED) surfaces to provide the most accurate representation of
the \hp\ global potential to date, overcoming the limitations on previous high
accuracy H$_3^+$ PESs near and above dissociation. PES75K+ can be used to
provide predictions of bound rovibrational energy levels with an accuracy of
approaching 0.1~\cm. Calculation of rovibrational energy levels within PES75K+
suggests that the non-adiabatic correction remains a limiting factor. The PES
is also constructed to give the correct asymptotic limit making it suitable for
use in studies of the H$^+$\,+\,H$_2$ prototypical chemical reaction. An
improved dissociation energy for H$_3^+$ is derived as
$D_0\,=\,$35\,076\,$\pm\,2\,$cm$^{-1}$.
