Using stimulated Raman adiabatic passage (SARP), it is possible, in principle, to transfer all the population in a rovibrational level of an isolated diatomic molecule to an excited rovibrational level. We use an overlapping sequence of pump (532 nm) and dump (683 nm) single-mode laser pulses of unequal fluence to prepare isolated H2 molecules in a molecular beam. In a first series of experiments we were able to transfer more than half the population to an excited rovibrational level [N. Mukherjee, W. R. Dong, J. A. Harrison, and R. N. Zare, J. Chem. Phys. 138(5), 051101-1-051101-4 (2013)]. Since then, we have achieved almost complete transfer (97% ± 7%) of population from the H2 (v = 0, J = 0) ground rovibrational level to the H2 (v = 1, J = 0) excited rovibrational level. An explanation is presented of the SARP process and how these results are obtained.