We have obtained spectra of the high-lying vibrational levels of the 13Σ+g state of Li26 via photoassociation of ultracold Li6 atoms confined in a magneto-optical trap. The 13Σ+g state of the diatomic molecule correlates to a 2S1/2 state atom plus a 2P1/2 state atom. The long-range part of the molecular interaction potential for this state depends on the 2P atomic radiative lifetime. By calculating the energy eigenvalues of a model potential for the 13Σ+g state and fitting them to the experimentally measured vibrational levels, we have extracted a value for the 2P lifetime of 26.99±0.16 ns. The precision is currently limited by the accuracy of a region of the model potential provided by ab initio calculations.