The Standard Model of particle physics is known to be incomplete. Extensions to the Standard Model, such as weak-scale supersymmetry, posit the existence of new particles and interactions that are asymmetric under time reversal (T) and nearly always predict a small yet potentially measurable electron electric dipole moment (EDM), d(e), in the range of 10(-27) to 10(-30) e.cm. The EDM is an asymmetric charge distribution along the electron spin ((S) over right arrow) that is also asymmetric under T. Using the polar molecule thorium monoxide, we measured d(e) = (-2.1 +/- 3.7(stat) +/- 2.5(syst)) x 10(-29) e.cm. This corresponds to an upper limit of vertical bar d(e)vertical bar < 8.7 x 10(-29) e.cm with 90% confidence, an order of magnitude improvement in sensitivity relative to the previous best limit. Our result constrains T-violating physics at the TeV energy scale.