A near-exact analytical solution of the statistical Susceptible-Infectious-Recovered (SIR) epidemics model for a constant ratio k0 of infection to recovery rates is derived. The derived solution is not of inverse form as the known solutions in the literature but expresses rather directly the three compartmental fractions S(τ), I(τ) and R(τ) and thus the rate of new infections j(τ) = S(τ)I(τ) in terms of the single function U(τ) and the reduced time τ (the time-integrated infection rate), involving the principal and non-principal branches of Lambert’s function. Exact analytical formulas for the peak time and the maximum fraction of I(τ) are obtained proving that the rate of new infections peaks before the fraction of infected persons. Our analysis is not entirely analytically exact because the reduced time dependence of the function U(τ) obeying a nonlinear integro-differential equation is only obtained approximately by expanding a double-exponential function to first-order at small reduced times, and employing an accurate simple approximation of the principal Lambert function at large times, respectively.