We present results of a large area CCD survey for low surface brightness galaxies (LSBs) that reaches central surface brightnesses of mu_0=25 mag/sqr-" in V. We have analyzed 17.5 degrees of transit scan data, and identified a statistical subset of 7 pure disk LSB's with mu_0 fainter than 23V-mag/sqr-" and with angular exponential scale lengths larger than alpha=2.5". The LSB detection is entirely automated, and the selection efficiency of the survey is well-quantified. After correcting for the selection efficiency, we find a surface density of 4.1^{+2.6}_{-2.1} galaxies/deg^2 for LSBs in the considered range of mu_0 and alpha (90% confidence levels). The final sample of LSBs are at distances comparable to those probed by large galaxy catalogs, and have intrinsic scale lengths of 1.7-3.6 kpc (H=50km/s/Mpc), based upon optical redshifts. We use the redshifts and the selection efficiency to calculate the number density in LSBs with 23<mu_0<25V-mag/sqr-" and find N = 0.01^{+0.006}_{-0.005} galaxies/Mpc^3, with 90% confidence. The measurement of N probably represents a lower limit, due to very strong biases against LSBs with bulges or edge-on LSBs in our sample. Comparing the LSB number density to the number density of normal galaxies with either similar scale lengths or similar luminosities, we find that the number density of LSBs with 23<mu_0<25V-mag/sqr-" is comparable to or greater than the number density of normal galaxies. The luminosity density in LSBs is comparable to the luminosity density of normal galaxies with similar luminosities, but is a factor of 3-10 smaller than the luminosity density of normal galaxies with similar scale lengths (ABRIDGED).Comment: Latex file, 41 pages total, including 13 pages of figures, uses aaspp4.sty & psfig.sty. Accepted to the Astronomical Journa