We present new measurements of the central stellar velocity dispersions (sigma) of dwarf elliptical galaxies (dE's), most of them of the nucleated variety. The data consist of photon counting spectra taken at the MMT with spectral resolution of 12 and 32 km s-1 and CCD data from CTIO and KPNO at 8 and 18 km s-1 (FWHM). The measured dispersions range from 14 to 39 km-1, and are lower than previous measurements done at more coarse resolution by about 50%. M/L ratios are derived for the galaxies. These are not unusually high (1-7, in solar units), but the change in M/L with luminosity for all dE galaxies confirms a prediction made by Dekel and Silk [ApJ, 303, 39 (1986)] (M/L approximately L-0.40+/-0.06) as does the observed relation between L and velocity dispersion (L approximately sigma5.6+/-0.9). We use the data to investigate the fundamental plane for dE's. A limited sample for which complete data exist (L, R(e), S(e), and sigma) indicates the galaxies are a one parameter family, i.e., a single parameter accurately predicts the three other structural parameters. However, a larger photometric data set (for which a are mostly unknown) indicates that the dwarfs require two parameters, mainly to explain the large low surface brightness galaxies. We attribute that poor correlation of color with structural parameters to real variations in the stellar population of dE's.