Low-resolution spectra of the close white dwarf-red dwarf binary RE J1629+780 are presented. The spectra, taken over the range 3400-8900A, show Balmer absorption features from the white dwarf and TiO bands and NaI absorption lines from the red dwarf. Also present in the spectra are narrow Balmer and CaII K lines in emission, believed to originate from the secondary star, probably as a result of reprocessed light from the hot white dwarf and/or the presence of active regions. From the TiO band ratio we estimate the spectral type of the red dwarf as Boeshaar dM4. Using pure hydrogen model fits to the Hβ, Hγ, Hdelta and Hepsilon Balmer absorption features, we derive a temperature of 41800+/-2000K and a surface gravity of log g=8.0+/-0.5 for the DA white dwarf, from which we obtain a mass of M_1=0.65+/-0.25 M_solar. We discuss the implications of our results and carry out a preliminary evolutionary analysis.