A method is developed for predicting the elasticity of a polymer melt through detailed atomistic simulations. The Helmholtz energy of a melt oriented by flow is postulated to be of the form A(T, rho, (c) over tilde), where T is the temperature, rho is the mass density, and (c) over tilde, the conformation tensor, is defined as the end-to-end tensor reduced by one-third the mean squared unperturbed end-to-end distance and averaged over all chains. The conjugate thermodynamic variable to (c) over tilde, alpha, is a tensorial orienting field intimately related to the strain rate in a flow situation. Assuming affine deformation of chain ends, the stress tensor tau can be expressed in terms of (c) over tilde and alpha. We have mapped out (c) over tilde, A, and tau for melts subjected to elongational flow by conducting Monte Carlo (MC) simulations at various values of alpha(xx), all other components of alpha being zero. Two linear polyethylene melts, of mean chain lengths C-24 and C-78 ia and polydispersity index 1.09, have been studied. Efficient sampling of oriented melt configurations has been made possible through the use of the end-bridging MC algorithm. Comparison of the melt, response to that of isolated chains subjected to the same orienting field shows that, while at low fields the two responses are similar, at high fields more anisotropy develops in the melt due to favorable lateral interactions between the oriented chains. Comparison against simple models used in flow calculations shows that FENE dumbbells and freely-jointed chains are more representative of the actual melt response than Hookean dumbbells, because they account for the finite extensibility of the polymer. Partitioning A into its energetic and entropic components shows that the melt response is purely entropic for long chains and low orienting fields, which leave the intrinsic shape of chains (averaged in the coordinate frame of their principal axes) practically unaltered. A significant energetic contribution develops for small chains and high orienting fields, where the chain intrinsic shape becomes more elongated and attractive lateral interchain interactions are intensified. Values of tau calculated from (c) over tilde and alpha are consistent with virial theorem predictions.