We report a measurement of the parity-violating asymmetry in fixed target electron-electron (Møller) scattering: A(PV)=[-175+/-30(stat)+/-20(syst)] x 10(-9). This first direct observation of parity nonconservation in Møller scattering leads to a measurement of the electron's weak charge at low energy Q(e)(W)=-0.053+/-0.011. This is consistent with the standard model expectation at the current level of precision: sin((2)theta(W)(M(Z))((-)MS)=0.2293+/-0.0024(stat)+/-0.0016(syst)+/-0.0006(theory).