The filoviruses, Marburg and Ebola, are non-segmented negative-strand RNA viruses causing severe hemorrhagic fever with high mortality rates in humans and nonhuman primates. The sequence of events that leads to release of filovirus particles from cells is poorly understood. Two contrasting mechanisms have been proposed, one proceeding via a "submarine-like" budding with the helical nucleocapsid emerging parallel to the plasma membrane, and the other via perpendicular "rocket-like" protrusion. Here we have infected cells with Marburg virus under BSL-4 containment conditions, and reconstructed the sequence of steps in the budding process in three dimensions using electron tomography of plastic-embedded cells. We find that highly infectious filamentous particles are released at early stages in infection. Budding proceeds via lateral association of intracellular nucleocapsid along its whole length with the plasma membrane, followed by rapid envelopment initiated at one end of the nucleocapsid, leading to a protruding intermediate. Scission results in local membrane instability at the rear of the virus. After prolonged infection, increased vesiculation of the plasma membrane correlates with changes in shape and infectivity of released viruses. Our observations demonstrate a cellular determinant of virus shape. They reconcile the contrasting models of filovirus budding and allow us to describe the sequence of events taking place during budding and release of Marburg virus. We propose that this represents a general sequence of events also followed by other filamentous and rod-shaped viruses.