We present a new technique to create a bin-averaged Hubble diagram (HD) from photometrically identified SN Ia data. The resulting HD is corrected for selection biases and contamination from core-collapse (CC) SNe, and can be used to infer cosmological parameters. This method, called "BEAMS with Bias Corrections" (BBC), includes two fitting stages. The first BBC fitting stage uses a posterior distribution that includes multiple SN likelihoods, a Monte Carlo simulation to bias-correct the fitted SALT-II parameters, and CC probabilities determined from a machine-learning technique. The BBC fit determines (1) a bin-averaged HD (average distance versus redshift), and (2) the nuisance parameters and β, which multiply the stretch and color (respectively) to standardize the SN brightness. In the second stage, the bin-averaged HD is fit to a cosmological model where priors can be imposed. We perform high-precision tests of the BBC method by simulating large (150,000 event) data samples corresponding to the Dark Energy Survey Supernova Program. Our tests include three models of intrinsic scatter, each with two different CC rates. In the BBC fit, the SALT-II nuisance parameters and β are recovered to within 1% of their true values. In the cosmology fit, we determine the dark energy equation of state parameter w using a fixed value of as a prior: averaging over all six tests based on 6 × 150,000 = 900,000 SNe, there is a small w-bias of . Finally, the BBC fitting code is publicly available in the SNANA package.