The thermally induced denaturation of DNA in the presence of an attractive solid surface is studied. The two strands of DNA are modeled via two coupled flexible chains without volume interactions. If the two strands are adsorbed on the surface, the denaturation phase transition disappears. Instead, there is a smooth crossover to a weakly naturated state. Our second conclusion is that even when the interstrand attraction alone is too weak for creating a naturated state at the given temperature and also when the surface-strand attraction alone is too weak for creating an adsorbed state, the combined effect of the two attractions can lead to a naturated and adsorbed state.