Place a droplet of mineral oil on water and the oil will spread to cover the water surface in a thin film. In this paper we study the everted problem: an aqueous droplet deposited onto a deep layer of silicone oil. As it is energetically favourable for the oil phase to spread to cover the droplet surface completely, the droplet is ultimately engulfed in the oil layer.We present a detailed study of engulfment dynamics, from the instant the droplet impacts the oil surface until it finally sediments into the less dense oil.We study a broad range of droplet sizes (micrometric to millimetric) and oil kinematic viscosities (102 to 105 cSt), corresponding to a viscosity-dominated parameter regime. We find that droplet engulfment involves the rapid submersion of the droplet driven by capillary forces in the oil surface, followed by the much slower peeling of the droplet from the interface, to which it is adhered via a thin cloaking layer of oil formed during the earlier stage. The later peeling stage is driven by a combination of geometric constraints at the apparent contact line and gravity pulling on the droplet. Gravitational effects are therefore essential to complete engulfment, even for micrometric droplets. Furthermore, the opposing effects of geometry and gravity result in the longest engulfment times for droplets of intermediate size. Experiments at fixed droplet size reveal a power law dependence of engulfment time on oil kinematic viscosity, which we argue reflects the dynamical formation of the oil cloaking layer.