  The electron neutrino mass has been measured in several tritium beta decay
experiments. These experiments are sensitive to a small neutrino mass because
the energy release of the decay is small. But the very smallness of the energy
release implies that the Coulomb interactions of the slowly moving emitted beta
electron are relatively large. Using field theoretic techniques, we derive a
systematic and controlled expansion which accounts for the Coulomb effects,
including the mutual interaction of the beta ray electron and the electron in
the final $^3{\rm He}^+$ ion. In our formulation, an effective potential which
describes the long range Coulomb force experienced by the beta ray is
introduced to ensure that our expansion is free of infrared divergences. Both
the exclusive differential decay rate to a specific final $^3{\rm He}^+$ state
and the inclusive differential decay rate are calculated to order $\eta^2$,
where $\eta$ is the usual Coulomb parameter. We analyze the order $\eta^2$
correction to the beta ray spectrum and estimate how it may affect the neutrino
mass squared parameter and the endpoint energy when this corrected spectrum is
used to compare with the experiments. We find that the effect is small.
