Colombi et al. (Paper I) investigated the counts-in-cells statistics and their respective errors in the τCDM Virgo Hubble Volume simulation. This extremely large N-body experiment also allows a numerical investigation of the cosmic distribution function, Υ( A^~), itself for the first time. For a statistic A, Υ( A^~) is the probability density of measuring the value A^~ in a finite galaxy catalogue. Υ was evaluated for the distribution of counts-in-cells, P_N, the factorial moments, F_k, and the cumulants, ψ^- and S_Ns, using the same subsamples as Paper I. While Paper I concentrated on the first two moments of Υ, i.e. the mean, the cosmic error and the cross-correlations, here the function Υ is studied in its full generality, including a preliminary analysis of joint distributions Υ( A^~, B^~). The most significant, and reassuring result for the analyses of future galaxy data is that the cosmic distribution function is nearly Gaussian provided its variance is small. A good practical criterion for the relative cosmic error is that ΔAA<~0.2. This means that for accurate measurements, the theory of the cosmic errors, presented by Szapudi & Colombi and Szapudi, Colombi & Bernardeau, and confirmed empirically by Paper I, is sufficient for a full statistical description and thus for a maximum likelihood rating of models. As the cosmic error increases, the cosmic distribution function Υ becomes increasingly skewed and is well described by a generalization of the lognormal distribution. The cosmic skewness is introduced as an additional free parameter. The deviation from Gaussianity of Υ( F^~_k) and Υ( S^~_N) increases with order k, N and similarly for Υ( P^~_N) when N is far from the maximum of P_N, or when the scale approaches the size of the catalogue. For our particular experiment, Υ( F^~_k) and Υ( ψ^-~) are well approximated with the standard lognormal distribution, as evidenced by both the distribution itself and the comparison of the measured skewness with that of the lognormal distribution.