The motion of interfaces for a mass-conserving Allen–Cahn equation that are attached to the boundary of a two-dimensional domain is studied. In the limit of thin interfaces, the interface motion for this problem is known to be governed by an area-preserving mean curvature flow. A numerical front-tracking method, that allows for a numerical solution of this type of curvature flow, is used to compute the motion of interfaces that are attached orthogonally to the boundary. Results obtained from these computations are favourably compared with a previously-derived asymptotic result for the motion of attached interfaces that enclose a small area. The area-preserving mean curvature flow predicts that a semi-circular interface is stationary when it is attached to a flat segment of the boundary. For this case, the interface motion is shown to be metastable and an explicit characterization of the metastability is given.