We present the highest quality Ly alpha forest spectra published to date. We have complete 7.9 km s(-1) FWHM spectra between the Ly alpha and Ly beta emission lines of the bright, high-redshift (V = 15.9, z = 3.05) QSO HS 1946 + 7658. The mean redshift of observed Ly alpha forest clouds is [z] = 2.7. The spectrum has a signal-to-noise ratio per pixel of 2 km s(-1) that varies from 15 at 4190 Angstrom to 100 at 4925 Angstrom. The absorption lines in the spectrum have been fitted with Voigt profiles, and the distribution of Voigt parameters has been analyzed. We show that fitting Voigt profiles to high quality data does not give unique results. We have performed simulations to differentiate between true features of the line distributions and artifacts of line blending and the fitting process. We show that the distribution of H I column densities is a power law of slope -1.5 from N(H I) = 10(14) cm(-2) to N(H I) < 10(12.1) cm(-2). We further show that our data is consistent with the hypothesis that this power law extends to N(H I) = 0, because lines weaker than N(H I) = 10(12.1) cm(-2) do not have a significant H I optical depth. At velocity dispersions between 20 and 60 km s(-1) the velocity dispersion (b) distribution is well described by a Gaussian with a mean of 23 km s(-1), and a sigma(b) of 14 km s(-1). Very similar N(H I) and b distributions were found at [z] = 3.7 by Lu et al. (1997), indicating no strong redshift evolution in these distributions for the Ly alpha forest. However, our b distribution has a lower mean and a wider dispersion than in past studies at the same redshift (e.g., Hu et al. 1995), which had lower signal-to-noise spectra. We unambiguously see narrow Ly alpha forest clouds with 14 km s(-1) less than or equal to b less than or equal to 20 km s(-1) that cannot be accounted for by noise effects. Our data also has absorption lines with b greater than or equal to 80 km s(-1) that cannot be explained by the blending of lower b lines. We find that the lower cutoff in the b distribution varies with N(H I), from b = 14 km s(-1) at N(H I) = 10(12.5) cm(-2) to b = 22 km s(-1) at N(H I) = 10(14.0) cm(-2). However, we see no similarly strong indication of a general correlation between b and N(H I). In contrast with previous results, we find no indication of Ly alpha forest line clustering on any scale above 50 km s(-1). Even among lines with 10(13.6) < N(H I) < 10(14.3) cm(-2), which were previously thought to cluster very strongly on velocity scales between 50 and 150 km s(-1), we see no clustering on any scale above 50 km s(-1), although we do see a 3 sigma clustering signal between 25 and 50 km s(-1) among these higher column density lines. With the distributions we have derived, we have calculated the expected He II optical depth of the Ly alpha forest. If there are no lines with N(H I) < 10(12.1) cm(-2), the Ly alpha forest is unlikely to provide a significant portion of the He II optical depth at high redshift. However, if the distribution extends to N(H I) < 10(9) cm(-2), the Ly alpha forest can provide all of the observed optical depth if N(He II)/N(H I) approximate to 100. We have calculated the redshift evolution of the optical depth from the He II Ly alpha forest based upon the line distributions we have derived for the H I Ly alpha forest. If the Ly alpha forest is responsible for the high-redshift He II optical depth and the spectral shape of the UV background does not change with redshift, we predict tau(He II) approximate to 2.4 at z = 3.3 to be consistent with the value of tau(He II) previously found at z = 2.4, provided that Ly alpha forest lines with N(H I) < 10(13) cm(-2) evolve like those with N(H I) > 10(13) cm(-2).
