Structural properties of a dense polymer melt confined between two hard walls are investigated over a wide range of temperatures by dynamic Monte Carlo simulation using the bond-fluctuation lattice model. The temperatures studied vary between those of the ordinary liquid and those of the strongly supercooled melt. At high temperatures the configurational properties of the polymers close to the walls exhibit the following features : The density profiles of the monomers and of the end monomers are enhanced at the walls and decay toward the bulk value on the length scale of the bond. Both the bonds and the chains tend to align preferentially parallel to the wall. The radius of gyration is the largest length scale characterizing the extent of the interfacial region. When the melt is progressively supercooled, all features of the profiles and the tendency of parallel orientation with respect to the walls become more pronounced. For the strongly supercooled melt the perturbation introduced by the presence of the walls penetrates deeply into the bulk, decaying on a length scale which is larger than the radius of gyration.