We report on deep observations of the extended TeV gamma-ray source MGRO J1908+06 made with the VERITAS very high energy ( VHE ) gamma-ray observatory . Previously , the TeV emission has been attributed to the pulsar wind nebula ( PWN ) of the Fermi -LAT pulsar PSR J1907+0602 . We detect MGRO J1908+06 at a significance level of 14 standard deviations ( 14 \sigma ) and measure a photon index of 2.20 \pm 0.10 _ { stat } \pm 0.20 _ { sys } . The TeV emission is extended , covering the region near PSR J1907+0602 and also extending towards SNR G40.5–0.5 . When fitted with a 2-dimensional Gaussian , the intrinsic extension has a standard deviation of \sigma _ { src } = 0.44 ^ { \circ } \pm 0.02 ^ { \circ } . In contrast to other TeV PWNe of similar age in which the TeV spectrum softens with distance from the pulsar , the TeV spectrum measured near the pulsar location is consistent with that measured at a position near the rim of G40.5–0.5 , 0.33 \arcdeg away .