Bilayer graphene is an attractive platform for studying new two-dimensional electron physics(1-5), because its flat energy bands are sensitive to out-of-plane electric fields and these bands magnify electron-electron interaction effects. Theory(6-16) predicts a variety of interesting broken symmetry states when the electron density is at the carrier neutrality point, and some of these states are characterized by spontaneous mass gaps, which lead to insulating behaviour. These proposed gaps(6,7,10) are analogous(17,18) to the masses generated by broken symmetries in particle physics, and they give rise to large Berry phase effects(8,19) accompanied by spontaneous quantum Hall effects(7-9,20). Although recent experiments(21-25) have provided evidence for strong electronic correlations near the charge neutrality point, the presence of gaps remains controversial. Here, we report transport measurements in ultra-clean double-gated bilayer graphene and use source-drain bias as a spectroscopic tool to resolve a gap of similar to 2 meV at the charge neutrality point. The gap can be closed by a perpendicular electric field of strength similar to 15 mV nm(-1), but it increases monotonically with magnetic field, with an apparent particle-hole asymmetry above the gap. These data represent the first spectroscopic mapping of the ground states in bilayer graphene in the presence of both electric and magnetic fields.