This paper presents a nonequilibrium equality for free energy differences. The result is an equality that allows one to extract equilibrium information (the free energy difference ΔF) from an ensemble of non-equilibrium (finite-time) measurements. The equality is: $$ \overline{\exp - \beta W} = \exp - \beta \Delta F, $$ or equivalently, $$ \Delta F = -\beta^{-1} \ln \overline{\exp - \beta W}, $$ where β ≡ 1/k_B T. This result is independent of the path γ from A to B and the rate at which the parameters are switched along the path. It is valid for sufficiently weak coupling between the system and the reservoir, and when quantal effects are negligible. The derivation of this result relies on the properties of Hamilton's equations and the assumption of weak coupling between the system and the reservoir. The result is shown to be valid for both isolated systems and systems coupled to a heat reservoir. It is also shown to be valid for numerical simulations using a Nosé-Hoover thermostat and the Metropolis Monte Carlo algorithm. The result has important implications for free energy calculations. It allows one to compute the exact value of ΔF by taking the average of exp - βW, then taking the logarithm and multiplying by -β^{-1}. This is in contrast to the usual approach, which uses the average of W as an upper bound on ΔF. The result is also shown to be valid in the limit of infinitely slow and infinitely fast switching of the external parameters. In the slow limit, the system is in quasi-static equilibrium with the reservoir, and the result reduces to the well-known identity: $$ \Delta F = \int_{0}^{1} d\lambda \langle \frac{\partial H_{\lambda}}{\partial \lambda} \rangle_{\lambda}. $$ In the fast limit, the result reduces to the well-known identity: $$ \Delta F = -\beta^{-1} \ln \langle \exp - \beta \Delta H \rangle_{0}. $$ These two results are well-established identities for the free energy difference ΔF. The result is also shown to be valid for systems coupled to a heat reservoir, and for systems simulated using a Nosé-Hoover thermostat or the Metropolis Monte Carlo algorithm. The result has important implications for the computation of free energy differences. It allows one to compute the exact value of ΔF by taking the average of exp - βW, then taking the logarithm and multiplying by -β^{-1}. This is in contrast to the usual approach, which uses the average of W as an upper bound on ΔF. The result is also shown to be valid for systems with a moderate number of degrees of freedom, such as nanoscale systems.This paper presents a nonequilibrium equality for free energy differences. The result is an equality that allows one to extract equilibrium information (the free energy difference ΔF) from an ensemble of non-equilibrium (finite-time) measurements. The equality is: $$ \overline{\exp - \beta W} = \exp - \beta \Delta F, $$ or equivalently, $$ \Delta F = -\beta^{-1} \ln \overline{\exp - \beta W}, $$ where β ≡ 1/k_B T. This result is independent of the path γ from A to B and the rate at which the parameters are switched along the path. It is valid for sufficiently weak coupling between the system and the reservoir, and when quantal effects are negligible. The derivation of this result relies on the properties of Hamilton's equations and the assumption of weak coupling between the system and the reservoir. The result is shown to be valid for both isolated systems and systems coupled to a heat reservoir. It is also shown to be valid for numerical simulations using a Nosé-Hoover thermostat and the Metropolis Monte Carlo algorithm. The result has important implications for free energy calculations. It allows one to compute the exact value of ΔF by taking the average of exp - βW, then taking the logarithm and multiplying by -β^{-1}. This is in contrast to the usual approach, which uses the average of W as an upper bound on ΔF. The result is also shown to be valid in the limit of infinitely slow and infinitely fast switching of the external parameters. In the slow limit, the system is in quasi-static equilibrium with the reservoir, and the result reduces to the well-known identity: $$ \Delta F = \int_{0}^{1} d\lambda \langle \frac{\partial H_{\lambda}}{\partial \lambda} \rangle_{\lambda}. $$ In the fast limit, the result reduces to the well-known identity: $$ \Delta F = -\beta^{-1} \ln \langle \exp - \beta \Delta H \rangle_{0}. $$ These two results are well-established identities for the free energy difference ΔF. The result is also shown to be valid for systems coupled to a heat reservoir, and for systems simulated using a Nosé-Hoover thermostat or the Metropolis Monte Carlo algorithm. The result has important implications for the computation of free energy differences. It allows one to compute the exact value of ΔF by taking the average of exp - βW, then taking the logarithm and multiplying by -β^{-1}. This is in contrast to the usual approach, which uses the average of W as an upper bound on ΔF. The result is also shown to be valid for systems with a moderate number of degrees of freedom, such as nanoscale systems.