This is the first reported use of a hybrid method involving density functional theory (DFT) and symmetry-adapted perturbation theory (SAPT) to calculate intermolecular interactions. This work was stimulated by the reported failures of supermolecular DFT calculations to adequately predict intermolecular land interatomic) interactions, particularly of the van der Waals type. The goals are to develop a hybrid scheme that will calculate intermolecular interaction energies accurately and in a computationally efficient fashion, while including the benefits of the energy decomposition provided by SAPT. The computational savings result from replacing the costly perturbation theory treatment with DFT, which should include the intramolecular correlation effects on the intermolecular interaction energies. The accuracy of this new hybrid approach (labeled SAPT(DFT)) is evaluated by comparisons with higher level calculations. The test cases include He-2, Ar-2. Ar-H-2, (H2O)(2), (HF)(2), CO2-CH3CN, and CO2-dimethylnitramine. The new approach shows mixed results concerning the accuracy of interaction energies. SAPT(DFT) correctly predicts all the qualitative trends in binding energies for all test cases. This is particularly encouraging in dimer systems dominated by dispersive interactions where supermolecular DFT fails to predict binding. In addition, the method achieves a drastic reduction (a factor of at least 100) in computational time over the higher level calculations often used to predict these forces. With respect to quantitative accuracy, this initial hybrid scheme, using the very popular exchange-correlation functional B3LYP, overestimates the second-order energy components (e.g., induction and dispersion terms) for all of the test cases, and subsequently overestimates the total interaction energy for all dimer systems except those heavily dominated by the electrotstatic interactions. The SAPT energy decomposition points to the use of DFT virtual orbital eigenvalues in the second-order perturbation terms as the likely cause for this error. These results are consistent with earlier work suggesting that DFT canonical virtual orbital energies obtained from commonly used functionals are less than optimal for use in such a perturbative scheme. The first-order interaction energy terms from the SAPT(DFT) are found to be generally more accurate than the second-order terms, and agree well with the benchmark values for dimers containing molecules with a permanent electric dipole moment. These first-order terms depend only upon the occupied MO eigenvectors, and hence are not affected by the inaccuracies in the Kohn-Sham DFT virtual orbital eigenvalues. These observations encourage future studies utilizing newly reported functionals, some of which have been developed to directly address problems with DFT virtual orbital energies and the asymptotic region of the electron density.