In a low-disorder two-dimensional electron system, when two Landau levels of opposite spin or pseudospin cross at the Fermi level, the dominance of the exchange energy can lead to a ferromagnetic, quantum Hall ground state whose gap is determined by the exchange energy and has skyrmions as its excitations. This is normally achieved via applying either hydrostatic pressure or uniaxial strain. We study here a very high-quality, low-density, two-dimensional hole system, confined to a 30-nm-wide (001) GaAs quantum well, in which the two lowest-energy Landau levels can be gate tuned to cross at and near filling factor . As we tune the field position of the crossing from one side of to the other by changing the hole density, the energy gap for the quantum Hall state at remains exceptionally large, and only shows a small dip near the crossing. The gap overall follows a dependence, expected for the exchange energy. Our data are consistent with a robust quantum Hall ferromagnet as the ground state.