First measurements of the W -> lv and Z/gamma* -> ll (l = e; mu) production cross sections in proton-proton collisions at root s = 7 TeV are presented using data recorded by the ATLAS experiment at the LHC. The results are based on 2250 W -> lv and 179 Z/gamma* -> ll candidate events selected from a data set corresponding to an integrated luminosity of approximately 320 nb(-1). The measured total W and Z/gamma*-boson production cross sections times the respective leptonic branching ratios for the combined electron and muon channels are sigma(tot)(W) center dot BR(W -> lv) = 9.96 +/- 0.23(stat) +/- 0.50(syst) +/- 1.10(lumi) nb and sigma(tot)(Z) center dot BR(Z/gamma* -> ll) = 0.82 +/- 0.06 (stat) +/- 0.05 (syst) +/- 0.09 (lumi) nb (within the invariant mass window 66 < m(ll) < 116 GeV). The W/Z cross-section ratio is measured to be 11.7 +/- 0.9(stat) +/- 0.4(syst). In addition, measurements of the W+ and W- production cross sections and of the lepton charge asymmetry are reported. Theoretical predictions based on NNLO QCD calculations are found to agree with the measurements.