The classical problem of water wave scattering by an infinite step in deep water with a free surface is extended here with an ice-cover modelled as a thin uniform elastic plate. The step exists between regions of finite and infinite depths and waves are incident either from the infinite or from the finite depth water region. Each problem is reduced to an integral equation involving the horizontal component of velocity across the cut above the step. The integral equation is solved numerically using the Galerkin approximation in terms of simple polynomial multiplied by an appropriate weight function whose form is dictated by the behaviour of the fluid velocity near the edge of the step. The reflection and transmission coefficients are obtained approximately and their numerical estimates are seen to satisfy the energy identity. These are also depicted graphically against thenon-dimensional frequency parameter for various ice-cover parameters in a number of figures. In the absence of ice-cover, the results for the free surface are recovered.