We investigate the shape of a growing interface in the presence of an impenetrable moving membrane. The two distinct geometrical arrangements of the interface and membrane, obtained by placing the membrane behind or ahead of the interface, are not symmetrically related. On the basis of numerical results and an exact calculation, we argue that these two arrangements represent two distinct universality classes for interfacial growth: while the well-established Kardar-Parisi-Zhang (KPZ) growth is obtained in the “ahead” arrangement, we find an arrested KPZ growth with a smaller roughness exponent in the “behind” arrangement. This suggests that the surface properties of growing cell membranes and expanding bacterial colonies, for example, are fundamentally distinct.