We consider the surface of a nearly incompressible polymer melt, extending the usual ground-state analysis of self-consistent field theory to describe finite length polymers in the ground-state potential. To maintain self-consistency, further corrections to the potential are calculated within linear response theory. From this, we find an excess of ends near the surface, followed by a compensating depletion on the R(g) length scale, which relies crucially on the finite compressibility of the melt. The attraction of ends to the surface can be described as resulting from a surface potential for ends with a strength on the order of k(B)T. Our results address the long-standing controversies of the distribution of chain ends, the chain-length dependence of the surface tension, and the interaction between objects immersed in a melt.