The covariant dynamics of a single classical electron subjected to the electromagnetic field of an intense, linearly polarized, ultrashort laser pulse in vacuum is studied in the context of the Dirac-Lorentz equation, which has long been suggested as a possible theory which includes radiative reaction. The Dirac-Lorentz equation is integrated numerically using Dirac{close_quote}s suggestion that the acausal (no runaway) solution should produce physically interesting consequences. This appears to be the case in our Compton scattering calculations, although the exact meaning of the results is not clear. {copyright} {ital 1996 The American Physical Society.}