Oil flow rates and cumulative discharge from the BP Macondo Prospect well in the Gulf of Mexico are calculated using a physically based model along with wellhead pressures measured at the blowout preventer (BOP) over the 86-day period following the Deepwater Horizon accident. Parameters appearing in the model are determined empirically from pressures measured during well shut-in and from pressures and flow rates measured the preceding day. This methodology rigorously accounts for ill-characterized evolution of the marine riser, installation and removal of collection caps, and any erosion at the wellhead. The calculated initial flow rate is 67,100 stock-tank barrels per day (stbd), which decays to 54,400 stbd just prior to installation of the capping stack and subsequent shut-in. The calculated cumulative discharge is 5.4 million stock-tank barrels, of which 4.6 million barrels entered the Gulf. Quantifiable uncertainties in these values are -9.3% and +7.5%, yielding a likely total discharge in the range from 4.9 to 5.8 million barrels. Minimum and maximum credible values of this discharge are 4.6 and 6.2 million barrels. Alternative calculations using the reservoir and sea-floor pressures indicate that any erosion within the BOP had little affect on cumulative discharge.