A survey of the low-lying states of the MgB2 and MgBB molecules is made, and it is found that different levels of theory lead to different calculated ground states. By employing high-level, up to RCCSD(T)/augcc-pCVQZ, ab initio calculations, we conclude that the lowest energy structure is a T-shaped X(1)A(1), state, but that there is a a(3)B(1) state lying within 1. kcal mol(-1) higher in energy, with (3)Sigma(-) and (5)Sigma(-) MgBB states lying similar to12 kcal mol(-1) above that. The ground state is confirmed at the CASSCF+MRCI+Q level. We also calculate DeltaH(f)[MgB2(X(1)A(1))], obtaining a value of 194 +/- 5 kcal mol(-1).