Differential cross sections for gamma p -> K+ Lambda (1405) and gamma p -> K+ Sigma(0) (1385) reactions have been measured in the photon energy range from 1.5 to 2.4 GeV and the angular range of 0.8 < cos(Theta) < 1.0 for the K+ scattering angle in the center-of-mass system. This data is the first measurement of the Lambda (1405) photoproduction cross section. The line shapes of Lambda (1405) measured in Sigma(+) pi(-) and Sigma(-) pi(+) decay modes were different with each other, indicating a strong interference of the isospin 0 and 1 terms of the Sigma pi scattering amplitudes. The ratios of Lambda (1405) production to Sigma(0) (1385) production were measured in two photon energy ranges: near the production threshold (1.5 < E-gamma < 2.0 GeV) and far from it ( 2.0 < E. < 2.4 GeV). The observed ratio decreased in the higher photon energy region, which may suggest different production mechanisms and internal structures for these hyperon resonances.