We present a precise measurement of the left-right cross section asymmetry (ALR) for Z boson production by e+e- collisions. The measurement was performed at a center-of-mass energy of 91.26 GeV with the SLD detector at the SLAC Linear Collider (SLC). The luminosity-weighted average polarization of the SLC electron beam was (63.0 ± 1.1)%. Using a sample of 49 392 Z decays, we measure ALR to be 0.1628 ± 0.0071 (stat) ± 0.0028 (syst) which determines the effective weak mixing angle to be sin2θWeff=0.2292±0.0009(stat)±0.0004(syst).