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Virtually all measurable properties of a classical fluid may be determined from the expectation value of the phase-space density operator f{rpt) =Z~B{r-r~(t)) 6(p -p (t)), and the phase-space density correlation function (f (rpt) f (r' p' t')) -(f (rp t)) (f(r' p' t')), a matrix with indices (rpt). Systematic procedures for approximating this matrix, unhindered by secular effects, are always based on approximations to its inverse. For a weakly coupled fluid, the inverse can be expanded in powers of~, the ratio of potential to kinetic energy.The leading term in this expansion gives rise to a Vlasov equation for the phase-space correlation function. The next term is the first that includes collisions, and results in relaxation towards equilibrium.This paper is concerned with the detailed study of the resulting fundamental nontrivial approximation.It is not Markovian and is perfectly reversible. Although the approximation is complicated, it is tractable analytically in various limits, and numerically for all wavelengths and frequencies. In this paper, only the behavior in certain limits is evaluated. Particular attention is directed toward its contractionsthe density correlation function, which is measured by inelastic neutron and light scattering, and the momentum correlation function. Calculation of the former at long wavelengths corroborates the Landau-Placzek expression for light scattering, and therefore demonstrates that the kinetic equation predicts hydrodynamic behavior at long times. Since the correlation function is correct to order X, it has, in contrast to a solution to the Boltzmann equation, the correct long-wavelength velocity of sound, c =-(dp/dmn), & 3 k&T/m. It also predicts different transport coefficients than those deduced from a Boltzman equation. These include a nonvanishing bulk viscosity. The transport coefficients reduce to those derived from the Boltzmann equation at low densities. Some aspects of the short-time behavior are also discussed.