We present the results of ROSAT PSPC observations of 10 quasars. These objects are part of our ROSAT program to observe a complete sample of optically selected quasars. This sample includes all 23 quasars from the bright quasar survey with a redshift z less than or equal to 0.400 and a Galactic H I column density N-HI(Gal) < 1.9 x 10(20) cm(-2) These selection criteria, combined with the high sensitivity and improved energy resolution of the PSPC, allow us to determine the soft (similar to 0.2-2 keV) X-ray spectra of quasars with about an order of magnitude higher precision compared with earlier soft X-ray observations. The following main results are obtained: Strong correlations are suggested between the soft X-ray spectral slope alpha(x) and the following emission line parameters: H beta FWHM, L([O III]) , and the Fe II/H beta flux ratio. These correlations imply the following: (1) The quasar's environment is likely to be optically thin down to similar to 0.2 keV. (2) In most objects alpha(x) varies by less than similar to 10% on timescales shorter than a few years. (3) alpha(x) might be a useful absolute luminosity indicator in quasars. (4) The Galactic He I and H I column densities are well correlated. Most spectra are well characterized by a simple power law, with no evidence for either significant absorption excess or emission excess at low energies, to within similar to 30%. We find [alpha(x)] = -1.50 +/- 0.40, which is consistent with other ROSAT observations of quasars. However, this average is significantly steeper than suggested by earlier soft X-ray observations of the Einstein IPC. The 0.3 keV flux in our sample can be predicted to better than a factor of 2 once the 1.69 mu m flux is given. This implies that the X-ray variability power spectra of quasars flattens out between f similar to 10(-5) and f similar to 10(-8) Hz. A steep alpha(x) is mostly associated with a weak hard X-ray component, relative to the near-IR and optical emission, rather than a strong soft excess, and the scatter in the normalized 0.3 keV flux is significantly smaller than the scatter in the normalized 2 keV flux. This argues against either thin or thick accretion disks as the origin of the soft X-ray emission. Further possible implications of the results found here are briefly discussed.