We report a measurement of the E-mode polarization power spectrum of the cosmic microwave background (CMB) using 150 GHz data taken from July 2014 to December 2016 with the POLARBEAR experiment. We reach an effective polarization map noise level of $32,mumathrm{K}$-$mathrm{arcmin}$ across an observation area of 670 square degrees. We measure the EE power spectrum over the angular multipole range $500 leq ell <3000$, tracing the third to seventh acoustic peaks with high sensitivity. The statistical uncertainty on E-mode bandpowers is $sim 2.3 mu {rm K}^2$ at $ell sim 1000$ with a systematic uncertainty of 0.5$mu {rm K}^2$. The data are consistent with the standard $Lambda$CDM cosmological model with a probability-to-exceed of 0.38. We combine recent CMB E-mode measurements and make inferences about cosmological parameters in $Lambda$CDM as well as in extensions to $Lambda$CDM. Adding the ground-based CMB polarization measurements to the Planck dataset reduces the uncertainty on the Hubble constant by a factor of 1.2 to $H_0 = 67.20 pm 0.57 {rm km,s^{-1} ,Mpc^{-1}}$. When allowing the number of relativistic species ($N_{eff}$) to vary, we find $N_{eff} = 2.94 pm 0.16$, which is in good agreement with the standard value of 3.046. Instead allowing the primordial helium abundance ($Y_{He}$) to vary, the data favor $Y_{He} = 0.248 pm 0.012$. This is very close to the expectation of 0.2467 from Big Bang Nucleosynthesis. When varying both $Y_{He}$ and $N_{eff}$, we find $N_{eff} = 2.70 pm 0.26$ and $Y_{He} = 0.262 pm 0.015$.
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