We report a chemically driven membrane shape instability that triggers the ejection of a tubule growing exponentially toward a chemical source. The instability is initiated by a dilation of the exposed monolayer, which is coupled to the membrane spontaneous curvature and slowed down by intermonolayer friction. Our experiments are performed by local delivery of a basic pH solution to a giant vesicle. Quantitative fits of the data give an intermonolayer friction coefficient b approximately 2x10;{9} J s/m;{4}. The exponential growth of the tubule may be explained by a Marangoni stress yielding a pulling force proportional to its length.