We studied the effect of disorder on the superconducting properties of polycrystalline MgB2 by specific-heat measurements. In the pristine state, these measurements give a bulk confirmation of the presence of two superconducting gaps with 2Delta0/kBTc = 1.3 and 3.9 with nearly equal weights. The scattering introduced by irradiation suppresses Tc and tends to average the two gaps although less than predicted by theory. We also found that by a suitable irradiation process by fast neutrons, a substantial bulk increase of dHc2/dT at Tc can be obtained without sacrificing more than a few degrees in Tc. The upper critical field of the sample after irradiation exceeds 28 T at T rightarrow 0.