The mixed boundary–value problem arising in the study of scattering of two–dimensional time–harmonic surface–water waves by a discontinuity on the surface boundary conditions, separating the clean surface and an ice–covered surface, is solved completely in the case of an infinite depth of water. The main problem is reduced to that of solving a singular integral equation, of the Carleman type, over a semi–finite range and the explicit solution of the original problem is determined. Neat and computable expressions are derived for the two most important quantities, known as the reflection and transmission coefficients, occurring in such scattering problems and tables of numerical values of these quantities are presented for specific choices of a parameter modelling the ice cover. The absolute values of the reflection and transmission coefficients are presented graphically. The present method of solution of the boundary–value problem produces simple expressions for the principal unknowns of the problem at hand and thus provides an easily understandable alternative to the rather complicated Wiener–Hopf method used previously.