GRB 080607 has provided the first observational signatures of molecular absorption bands toward any galaxy hosting a gamma-ray burst (GRB). Despite the identification of dozens of features as belonging to various atomic and molecular (H$_2$ and CO) carriers, many more absorption features remained unidentified. Here, we report on a search among these features for absorption from vibrationally excited H$_2$, a species that was predicted to be produced by the UV flash of a GRB impinging on a molecular cloud. Following a detailed comparison between our spectroscopy and static, as well as dynamic, models of H$_2$* absorption, we conclude that a column density of 10$^17.5±0.2$ cm$^-2$ of H$_2$* was produced along the line of sight toward GRB 080607. Depending on the assumed amount of dust extinction between the molecular cloud and the GRB, the model distance between the two is found to be in the range 230-940 pc. Such a range is consistent with a conservative lower limit of 100 pc estimated from the presence of Mg I in the same data. These distances show that substantial molecular material is found within hundreds of pc from GRB 080607, part of the distribution of clouds within the GRB host galaxy.