We report a first, complete lattice QCD calculation of the long-distance contribution to the K + → π + νν decay within the standard model. This is a second-order weak process involving two four-Fermi operators that is highly sensitive to new physics and being studied by the NA62 experiment at CERN. While much of this decay comes from perturbative, short-distance physics there is a long-distance part, perhaps as large as the planned experimental error, which involves nonperturbative phenomena. The calculation presented here, with unphysical quark masses, demonstrates that this contribution can be computed using lattice methods by overcoming three technical difficulties: 1) a short-distance divergence that results when the two weak operators approach each other, 2) exponentially-growing, unphysical terms which appear in Euclidean, second-order perturbation theory and 3) potentially large finite-volume effects. A follow-on calculation with physical quark masses and controlled systematic errors will be possible with the next generation of computers.Keywords: rare kaon decay, lattice QCD Introduction -An important objective of experimental high-energy physics is the search for direct and indirect signs of new physics. Complementary to the direct search for new particles and forces at high energy, is the search for subtle deviations from standard model predictions at lower energies. The rare kaon decays K → πνν are such examples. As flavor-changing-neutral-current processes, the K → πνν decay amplitudes arise as oneloop, electroweak effects. The small size of one-loop, standard model effects makes these decays particularly sensitive to new phenomena.