The recent identification of one or two subparsec disks of young, massive stars orbiting the-4 x 106 M⊙ black hole Sgr A* has prompted an in situ scenario for star formation in disks of gas formed from a cloud captured from the Galactic center environment. To date there has been no explanation given for the low angular momentum of the disks relative to clouds passing close to the center. Here we show that the partial accretion of extended Galactic center clouds, such as the 50 km s-1 giant molecular cloud, that temporarily engulf Sgr A* during their passage through the central region of the Galaxy provide a natural explanation for the angular momentum and surface density of the observed stellar disks. The captured cloud material is gravitationally unstable and forms stars as it circularizes, potentially explaining the large eccentricity and range of inclinations of the observed stellar orbits. The application of this idea to the formation of the circumnuclear ring is also discussed.