The effects of the positron (e(+)) annihilation in e(+) scattering by the H atom are included directly in the Hamiltonian as an absorption potential, and hence the finite lifetimes Gamma(ns) of positronium (Ps) in states ns are automatically taken account of. The Schrödinger equation is solved using the hyperspherical close-coupling method. The annihilation and Ps formation are shown to be inseparable near the Ps(1s) threshold E(th); Ps formation constitutes just an indirect pathway to annihilation. The annihilation cross section, which would diverge near E(th) if Gamma(1s) were infinite, connects smoothly across E(th) to the cross section for Ps formation, which is meaningful only at energies well above E(th).