We present a three-dimensional (3-D) numerical ideal magnetohydrodynamics (MHD) model describing the time-dependent expulsion of a coronal mass ejection (CME) from the solar corona propagating to 1 astronomical unit (AU). The simulations are performed using the Block Adaptive Tree Solar-Wind Roe Upwind Scheme (BATS-R-US) code. We begin by developing a global steady-state model of the corona that possesses high-latitude coronal holes and a helmet streamer structure with a current sheet at the equator. The Archimedean spiral topology of the interplanetary magnetic field is reproduced along with fast and slow speed solar wind. Within this model system, we drive a CME to erupt by the introduction of a Gibson-Low magnetic flux rope that is anchored at both ends in the photosphere and embedded in the helmet streamer in an initial state of force imbalance. The flux rope rapidly expands and is ejected from the corona with maximum speeds in excess of 1000 km/s. Physics-based adaptive mesh refinement (AMR) allows us to capture the structure of the CME focused on a particular Sun-Earth line with high spatial resolution given to the bow shock ahead of the flux rope as well as to the current sheet behind. The CME produces a large magnetic cloud at 1 AU (>100 R⊙) in which Bz undergoes a full rotation from north to south with an amplitude of 20 nT. In a companion paper, we find that the CME is very effective in generating strong geomagnetic activity at the Earth in two ways. First, through the strong sustained southward Bz (lasting more than 10 hours) and, second, by a pressure increase associated with the CME-driven shock that compresses the magnetosphere.