A determination of the spin and parity of the Lambda(1405) is presented using photoproduction data from the CLAS detector at Jefferson Lab. The reaction gamma + p -> K+ + Lambda(1405) is analyzed in the decay channel Lambda(1405) -> Sigma(+) + pi(-), where the decay distribution to Sigma(+)pi(-) the variation of the Sigma(+) polarization with respect to the Lambda(1405) polarization direction determines the parity. The Lambda(1405) is produced, in the energy range 2.55 <W <2.85 GeV and for 0.6 <cos theta(c.m.)(K+) <0.9, with polarization P = 0.45 +/- 0.02(stat) +/- 0.07(syst). The analysis shows that the decays are in S wave, with the Sigma(+) polarized such that the Lambda(1405) has spin-parity J(P) = 1(-)/2, as expected by most theories.