We report on single molecule measurements of the viscoelastic properties of the polysaccharide dextran using a new approach which involves acquiring the power spectral density of the thermal noise of an atomic force microscope cantilever while holding the single molecule of interest under force-clamp conditions. The attractiveness of this approach when compared with techniques which use forced oscillations under constant loading rate conditions is that it is a near-equilibrium measure of mechanical response which provides a more relevant probe of thermally driven biomolecular dynamics. Using a simple harmonic oscillator model of the cantilever-molecule system and by subtracting the response of the free cantilever taking into account hydrodynamic effects, the effective damping zeta(mol) and elastic constant k(mol) of a single molecule are obtained. The molecular elasticity measured by this new technique shows a dependence on applied force that reflects the chair-boat conformational transition of the pyranose rings of the dextran molecule which is in good agreement with values obtained directly from the gradient of a conventional constant loading rate force-extension curve. The molecular damping is also seen to follow a nontrivial dependence on loading which we suggest indicates that it is internal friction and not work done on the solvent that is the dominant dissipative process.