We describe a new thermonuclear explosion model for Type I (or Type II) supernovae whereby relativistic terms enhance the self gravity of a carbon-oxygen white dwarf (or red-giant core) as it passes or orbits near a black hole. This relativistic compression can cause the central density to exceed the threshold for pycnonuclear or thermonuclear reactions so that an explosion ensues. We have considered three possible environments: 1) white dwarfs orbiting a low-mass (∼ 10 – 20 M ⊙ ) black hole; 2) white dwarfs encountering a massive (∼ 1 – 3 × 10 3 M ⊙ ) black hole in a dense globular cluster; and 3) white dwarfs passing a supermassive (∼ 10 6 – 10 9 M ⊙ ) black hole in a dense galactic core. We estimate the rate at which such events could occur to be significantly less than the rate of normal Type Ia supernovae for all three classes. Nevertheless, they should be frequent enough to warrant a search for this new class of supernova. We show results of three-dimensional thermonuclear burn calculations of white dwarfs or red-giant cores ignited near a supermassive black hole. Such an event might have produced the observed "mixed-morphology" Sgr A East supernova remnant (SNR) in the Galactic core.