We study the long-range attractive force between neutral sodium clusters Na-n (2 less than or equal to n less than or equal to 20) and the alkali atoms Li, Na, and K. Absolute cross sections for the scattering of a beam of clusters by atomic vapor are measured, and are shown to arise from the van der Waals dispersive interaction V = -C-6/r(6). These cross sections are extremely large (up to thousands of Angstrom(2) in the center-of-mass frame). Their magnitudes an in good agreement with predictions based on the London theory of dispersion forces; the large strength of the interaction is a consequence of the high cluster polarizabilities. In addition, we evaluate the contribution of the higher-order potential term -C-8/r(8) to the scattering cross section and show that it can become quite significant for collisions involving large clusters (n similar to 10(2)-10(3)).