Conformational properties of polymer melts confined between two hard structureless walls are investigated by Monte Carlo simulation of the bond-fluctuation model. Parallel and perpendicular components of chain extension, bond-bond correlation function and structure factor are computed and compared with recent theoretical approaches attempting to go beyond Flory's and Silberberg's hypotheses. We demonstrate that for ultrathin films where the thickness, $H$, is smaller than the excluded volume screening length (blob size), $\xi$, the chain size parallel to the walls diverges logarithmically, $R^2/2N \approx b^2 + c \log(N)$ with $c \sim 1/H$. The corresponding bond-bond correlation function decreases like a power law, $C(s) = d/s^{\omega}$ with $s$ being the curvilinear distance between bonds and $\omega=1$. % Upon increasing the film thickness, $H$, we find -- in contrast to Flory's hypothesis -- the bulk exponent $\omega=3/2$ and, more importantly, an {\em decreasing} $d(H)$ that gives direct evidence for an {\em enhanced} self-interaction of chain segments reflected at the walls. Systematic deviations from the Kratky plateau as a function of $H$ are found for the single chain form factor parallel to the walls in agreement with the {\em non-monotonous} behaviour predicted by theory. This structure in the Kratky plateau might give rise to an erroneous estimation of the chain extension from scattering experiments. For large $H$ the deviations are linear with the wave vector, $q$, but are very weak. In contrast, for ultrathin films, $H<\xi$, very strong corrections are found (albeit logarithmic in $q$) suggesting a possible experimental verification of our results.