This article presents the exact solutions for the unsteady flow of an incompressible Burgers' fluid induced by the impulsive motion of a plate between two sidewalls perpendicular to the plate. Employing the Fourier sine transforms, the expressions for the velocity field, the tangential stresses, and the volume flux are obtained. The obtained solutions satisfy all imposed initial and boundary conditions and in the absence of the sidewalls reduce to the solutions corresponding to the flow over an infinite flat plate. The effect of the material parameters on the velocity field and the tangential stress at the bottom wall is spotlighted by means of the graphical illustrations.