Rotationally resolved S-1 <- S-0 electronic spectra of 1,2-dimethoxybenzene (DMB) and its water complex have been observed and assigned. The derived values of the rotational constants show that the bare molecule has a planar heavy-atom structure with trans-disposed methoxy groups in its ground and excited electronic states. The 0(0)(0) transition of DMB is polarized along the b-axis bisecting the methoxy groups, demonstrating that its S-1 state is an L-1(b) state. Higher energy bands of DMB are also polarized along the b-axis and have been tentatively assigned to different vibrational modes of the 1Lb state. The water complex origin appears 127 cm(-1) to the blue of the bare molecule origin. Analyses of the high resolution spectra of DMB/H2O and DMB/D2O suggest that the water molecule is attached via two O-(HO)-O-... hydrogen bonds to the methoxy groups in both electronic states. A tunneling motion of the attached water molecule is revealed by a splitting of these spectra into two subbands. Potential barriers to this motion have been determined.