Dispersion forces are long-range interactions between polarizable objects that arise from fluctuations in the electromagnetic field between them. Dispersion forces have been observed between microscopic objects such as atoms and molecules (the van der Waals interaction)(1), between macroscopic objects (the Casimir interaction)(2) and between an atom and a macroscopic object (the Casimir-Polder interaction)(2,3). Dispersion forces are known to increase the attractive forces between the components in nanomechanical devices(4), to influence adsorption rates onto nanostructures(5), and to influence the interactions between biomolecules in biological systems(1). In recent years, there has been growing interest in studying dispersion forces in nanoscale systems(6-8) and in exploring the interactions between carbon nanotubes and cold atoms(9-11). However, there are considerable difficulties in developing dispersion force theories for general, finite geometries such as nanostructures. Thus, there is a need for new experimental methods that are able to go beyond measurements of planar surfaces(12-15) and nanoscale gratings(16) and make measurements on isolated nanostructures. Here, we measure the dispersion force between a rubidium atom and a multiwalled carbon nanotube by inserting the nanotube into a cloud of ultracold rubidium atoms and monitoring the loss of atoms from the cloud as a function of time. We perform these experiments with both thermal clouds of ultracold atoms and with Bose-Einstein condensates. The results obtained with this approach will aid the development of theories describing quantum fields near nanostructures, and hybrid cold-atom/solid-state devices may also prove useful for applications in quantum sensing and quantum information(17,18).