The properties of silicon dioxide have been studied extensively over the years. However, there still remain major unanswered questions regarding the nature of the optical spectrum and the role of excitonic effects in this technologically important material. In this work, we present an ab initio study of the optical absorption spectrum of alpha-quartz, using a newly developed first-principles method which includes self-energy and electron-hole interaction effects. The quasiparticle band structure is computed within the GW approximation to obtain a quantitative description of the single-particle excitations. The Bethe-Salpeter equation for the electron-hole excitations is solved to obtain the optical spectrum and to understand the spatial extent and physical properties of the excitons. The theoretical absorption spectrum is found to be in excellent agreement with the measured spectrum. We show that excitonic effects are crucial in the frequency range up to 5 eV above the absorption threshold.