In this paper, we report new limits on 21 cm emission from cosmic reionization based on a 135 day observing campaign with a 64-element deployment of the Donald C. Backer Precision Array for Probing the Epoch of Reionization in South Africa. This work extends the work presented in Parsons et al. with more collecting area, a longer observing period, improved redundancy-based calibration, improved fringe-rate filtering, and updated power-spectral analysis using optimal quadratic estimators. The result is a new 2σ upper limit on Δ 2 (k) of (22.4 mK) 2 in the range k h 0.15 0.5 Mpc 1 < < -at z = 8.4. This represents a three-fold improvement over the previous best upper limit. As we discuss in more depth in a forthcoming paper, this upper limit supports and extends previous evidence against extremely cold reionization scenarios. We conclude with a discussion of implications for future 21 cm reionization experiments, including the newly funded Hydrogen Epoch of Reionization Array.