  We simulate the Lya forest in a standard CDM universe using a 2-level
hierarchical grid code to evolve the dark and baryonic matter components
self-consistently. We solve the time-dependent ionization equations for
hydrogen and helium, adopting the Haardt & Madau (1996) estimate for the
metagalactic UV radiation background. We compare our simulation results with
the measured properties of the Lya forest by constructing synthetic spectra
which we analyze in a manner similar to the method used by observers. The HI
column density and Doppler parameter distributions we obtain agree closely with
those measured by the Keck HIRES and earlier high spectral resolution
observations over the column density range 10^{12} < NHI < 10^{16}. In
particular, we find a power-law column density distribution persists to the
lowest values reported from the HIRES measurements, in agreement with the
incompleteness-corrected observations. We find evolution in the cloud number
density and opacity comparable to the observed evolution. We also compare with
measured values of the intergalactic HeII opacity. Our results require a HeII
ionizing background lower than the Haardt & Madau estimate by a factor of 4,
corresponding to a soft intrinsic QSO spectrum of alpha_Q = 1.8 - 2.
