We have successfully simulated the magnetosphere of Uranus for the time period of the Voyager 2 flyby in January 1986. On the basis of the Voyager measurements, a self-consistent numerical solution is obtained with the parallel block adaptive three-dimensional (3-D) MHD code BATS-R-US. The time-dependent simulation has been carried out with a new explicit-implicit time integration scheme. By comparing corotating steady state solutions and a fully time-dependent 3-D simulation with the Voyager data, we show that the magnetosphere of Uranus at the time of the flyby can be regarded as stationary relative to the frame corotating with the planet. We obtained excellent agreement with the observed magnetic field vector along the whole path of the flyby, which includes the near-Uranus offset dipole field as well as several current sheet crossings in the tail. The location of the bow shock and the magnetopause also agree to high accuracy. We are confident that our numerical solution is a good representation of the three-dimensional magnetosphere of Uranus during the flyby. The numerical solution shows a twisted magnetotail with field lines that are also stretched due to the flow of plasma in the magnetotail.