We study a Type Ia supernova explosion using large-scale three-dimensional numerical simulations based on reactive fluid dynamics with a simplified mechanism for nuclear reactions and energy release. The initial deflagration stage of the explosion involves a subsonic turbulent thermonuclear flame propagating in the gravitational field of an expanding white dwarf. The deflagration produces an inhomogeneous mixture of unburned carbon and oxygen with intermediate-mass and iron-group elements in central parts of the star. During the subsequent detonation stage, a supersonic detonation wave propagates through the material unburned by the deflagration. The total energy released in this delayed-detonation process, (1.3-1.6) × 10⁵¹ ergs, is consistent with a typical range of kinetic energies obtained from observations. In contrast to the deflagration model, which releases only about 0.6 × 10⁵¹ ergs, the delayed-detonation model does not leave carbon, oxygen, and intermediate-mass elements in central parts of a white dwarf. This removes the key disagreement between three-dimensional simulations and observations, and makes a delayed detonation the mostly likely mechanism for Type Ia supernova explosions.