When using appropriate surfactants, oil and aqueous foam can be intimately mixed without the foam being destroyed. In this Letter, we show that a foam, initially free of oil, can draw an oil drop under the action of capillary forces and stretch it through the aqueous network. We focus on the suction of oil by a single horizontal foam channel, known as a Plateau border. In such confined channels, imbibition dynamics are governed by a balance between capillarity and viscosity. Yet, the scaling law for our system differs from that of classical imbibition in porous media such as aqueous foam. This is due to the particular geometry of the liquid channels: Plateau borders filled with foaming solution are always concave whereas they can be convex or flat when filled with oil. Finally, the oil slug, confined in the Plateau border, fragments into droplets following a film breakup.