We describe a standard star catalog constructed using multiple SDSS photometric observations (at least four per band, with a median of 10) in the ugriz system. The catalog includes 1.01 million nonvariable unresolved objects from the equatorial stripe 82 (|dJ2000.0| < 1.266°) in the right ascension range 20h34m-4h00m and with the corresponding r-band (approximately Johnson V-band) magnitudes in the range 14-22. The distributions of measurements for individual sources demonstrate that the photometric pipeline correctly estimates random photometric errors, which are below 0.01 mag for stars brighter than 19.5, 20.5, 20.5, 20, and 18.5 in ugriz, respectively (about twice as good as for individual SDSS runs). Several independent tests of the internal consistency suggest that the spatial variation of photometric zero points is not larger than ~0.01 mag (rms). In addition to being the largest available data set with optical photometry internally consistent at the ~1% level, this catalog provides a practical definition of the SDSS photometric system. Using this catalog, we show that photometric zero points for SDSS observing runs can be calibrated within a nominal uncertainty of 2% even for data obtained through 1 mag thick clouds, and we demonstrate the existence of He and H white dwarf sequences using photometric data alone. Based on the properties of this catalog, we conclude that upcoming large-scale optical surveys such as the Large Synoptic Survey Telescope will be capable of delivering robust 1% photometry for billions of sources.