We present a deep optical survey of Uranus's Hill sphere for small satellites. The 8 m Subaru Telescope was used to survey about 3.5 square degrees with a 50% detection efficiency at limiting red magnitude mR=26.1. This magnitude corresponds to objects that are about 7 km in radius (assuming an albedo of 0.04). We detected (without prior knowledge of their positions) all previously known outer satellites and discovered two new irregular satellites (S/2001 U2 and S/2003 U3). The two inner satellites Titania and Oberon were also detected. One of the newly discovered bodies (S/2003 U3) is the first known irregular prograde satellite of the planet. The population, size distribution, and orbital parameters of Uranus's irregular satellites are remarkably similar to those of the irregular satellites of gas giant Jupiter. Both have shallow size distributions (power-law indices q~2 for radii larger than 7 km) with no correlation between the sizes of the satellites and their orbital parameters. However, unlike those of Jupiter, Uranus's irregular satellites do not appear to occupy tight, distinct dynamical groups in semimajor-axis versus inclination phase space. Two groupings in semimajor-axis versus eccentricity phase space appear to be statistically significant. Based largely on data collected at the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan. Based in part on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership.