The present paper addresses the problem of contact between a rigid hemisphere and a thin elastic layer strongly bonded on a rigid plane support, which can be thought of as an adhesive obeying a geometrically non-linear behaviour due to the change of contact area. Using the asymptotic expansion method from a three-dimensional analysis of the layer, a two-dimensional model is derived, under the assumptions of large displacements and small strains. The leading term of the solution of the asymptotic development is such that the displacement field varies linearly through the layer thickness and the stress tensor is constant. A quasi-linear relation is obtained between the area of contact and the penetration of the hemisphere within the layer, and the variation with penetration of the compressive load exerted by the hemisphere is seen to give satisfactory agreement with experiments. In the last part, we present theoretical results concerning the rupture behaviour; the effect of adhesion energy between the hemisphere and the layer on the radius of curvature at the rupture point between both solids is assessed. Further, the thickness of an hypothetical interphase through which failure propagates is determined theoretically.