We report a measurement of the B-mode polarization power spectrum in the cosmic microwave background (CMB) using the Polarbear experiment in Chile. The faint B-mode polarization signature carries information about the universe's entire history of gravitational structure formation, and the cosmic inflation that may have occurred in the very early universe. Our measurement covers the angular multipole range 500 < l < 2100 and is based on observations of an effective sky area of 25 deg2 with 3′.5 resolution at 150 GHz. On these angular scales, gravitational lensing of the CMB by intervening structure in the universe is expected to be the dominant source of B-mode polarization. Including both systematic and statistical uncertainties, the hypothesis of no B-mode polarization power from gravitational lensing is rejected at 97.2% confidence. The band powers are consistent with the standard cosmological model. Fitting a single lensing amplitude parameter ABB to the measured band powers, ABB = 1.12 ± 0.61(stat)-0.12 +0.04 (sys) ± 0.07(multi), where ABB = 1 is the fiducial wmap-9 ΛCDM value. In this expression, "stat" refers to the statistical uncertainty, "sys" to the systematic uncertainty associated with possible biases from the instrument and astrophysical foregrounds, and "multi" to the calibration uncertainties that have a multiplicative effect on the measured amplitude ABB.