Electron affinities of the alkali atoms sodium to eka-francium are calculated by the intermediate Hamiltonian Fock-space coupled cluster approach, which allows very large P spaces. Large basis sets are used (37s32p23d18f10g7h for most atoms), and many electrons are correlated (from 10 for Na- to 52 for E119(-)) to account for core polarization. While the usual Fock-space method gives errors of 5%-9% for K, Rb, and Cs, the intermediate Hamiltonian results agree with all known values to 5 meV or 1%. The EA of Fr, not known experimentally, is predicted at 491+/-5 meV. While EAs decrease from Li to Cs, the Fr value is 20 meV higher than that of Cs, with E119 EA being much higher at 662 meV. This trend reversal is due to relativistic stabilization of s orbitals, which has been shown [Eliav , Phys. Rev. Lett. 74, 1079 (1995)] to give the rare gas E118 positive electron affinity.