When bubbles are continuously released from a localised source at the bottom of a liquid layer, a plume is produced. As the bubble plume rises due to its buoyancy, it entrains surrounding liquid, which is carried upward with the stream of bubbles. In the present work, we investigate the motion of a plume of low-Reynolds number bubbles in a stratified liquid consisting of two homogeneous layers of different densities. The liquid environment is of finite lateral extent. We develop a theoretical model for the flow of the bubble plume and the surrounding liquid. The full equations are solved numerically. The mixing at the interface is quantified and the time-evolution of the density profiles in both layers is calculated. The model also predicts the rate of rise of the density interface. We develop an analytical solution For the problem in the limit of strong stratifications. Our theoretical predictions are compared with new experimental results using plumes of small bubbles generated by electrolysis of an aqueous solution of sodium chloride and with previous experimental results (McDougall (1978), Journal of Fluid Mechanics 85, 655-672; Baines & Leitch (1992), Journal of Hydraulic Engineering, 118(4), 559-577).