Analytic evaluation of the amplitudes for orthopositronium decay to three photons to one-loop order

The matrix element for the decay of orthopositronium to three photons can be expressed in terms of three independent amplitudes. We describe the analytic evaluation of these amplitudes, both to lowest order and with the inclusion of all one-loop corrections. We use these amplitudes to find precise values for the one-loop correction to the orthopositronium decay rate {gamma}{sub 1}=-10.286606(10)({alpha}/{pi}){gamma}{sub LO}, and for the order-{alpha}{sup 2} 'square' correction to the decay rate {gamma}{sub 2}(square)=28.860(2)({alpha}/{pi}){sup 2}{gamma}{sub LO}, where {gamma}{sub LO} is the lowest order rate. We give in explicit form the function describing the one-loop correction to the distribution in phase space of the final state photons.