The free energy of turning on interactions in a many-body Hamiltonian is critical to estimating the equilibrium thermodynamic stability of many molecular systems. Thermodynamic integration (TI) provides a rigorous means to estimating such a free-energy difference. Unfortunately, TI calculations are computationally expensive, due to the need to sample many interpolating ensembles with sufficient conformational-space overlap. In this work, we propose to use a denoising diffusion model (DDM) to perform TI solely from the two end points. Critically, we map the latent space of the DDM to the \emph{non-interacting} Hamiltonian. By parametrizing the score of the DDM as the gradient of a time-dependent energy function, we can exploit automatic differentiation to accurately and efficiently estimate TI. We apply our method to a box of Lennard-Jones particles with varying numbers of particles. The results show that we can accurately calculate grand-canonical distribution functions, illustrating the accuracy of the estimated free energies.