We present the results of our analysis of cosmic-ray electrons using about 8 x 10(6) electron candidates detected in the first 12 months on-orbit by the Fermi Large Area Telescope. This work extends our previously published cosmic-ray electron spectrum down to 7 GeV, giving a spectral range of approximately 2.5 decades up to 1 TeV. We describe in detail the analysis and its validation using beam-test and on-orbit data. In addition, we describe the spectrum measured via a subset of events selected for the best energy resolution as a cross-check on the measurement using the full event sample. Our electron spectrum can be described with a power law proportional to E-3.08+/-0.05 with no prominent spectral features within systematic uncertainties. Within the limits of our uncertainties, we can accommodate a slight spectral hardening at around 100 GeV and a slight softening above 500 GeV.