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We report a measurement of the E-mode polarization power spectrum of the cosmic microwave background (CMB) using 150 GHz data taken from 2014 July to 2016 December with the POLARBEAR experiment. We reach an effective polarization map noise level of 32 mK-arcmin across an observation area of 670 square degrees. We measure the EE power spectrum over the angular multipole range 500 ≤ ℓ < 3000, tracing the third to seventh acoustic peaks with high sensitivity. The statistical uncertainty on E-mode bandpowers is ∼2.3 μK2 at ℓ ∼ 1000, with a systematic uncertainty of 0.5 mK2. The data are consistent with the standard ΛCDM cosmological model with a probability-to-exceed of 0.38. We combine recent CMB E-mode measurements and make inferences about cosmological parameters in ΛCDM as well as in extensions to ΛCDM. Adding the ground-based CMB polarization measurements to the Planck data set reduces the uncertainty on the Hubble constant by a factor of 1.2 to H0 = 67.20 ±0.57 km s- Mpc- 1 1. When allowing the number of relativistic species (Neff ) to vary, we find Neff = 2.94 ±0.16, which is in good agreement with the standard value of 3.046. Instead allowing the primordial helium abundance (YHe) to vary, the data favor YHe = 0.248 ±0.012. This is very close to the expectation of 0.2467 from big bang nucleosynthesis. When varying both YHe and Neff , we find Neff = 2.70 ±0.26 and YHe = 0.262 ±0.015.