Motivated by recent Transition Region and Coronal Explorer (TRACE) observations of damped oscillations in coronal loops, we consider analytically the motion of an inhomogeneous coronal magnetic tube of radius a in a zero-β plasma. An initially perturbed tube may vibrate in its kink mode of oscillation, but those vibrations are damped. The damping is due to resonant absorption, acting in the inhomogeneous regions of the tube, which leads to a transfer of energy from the kink mode to Alfvén (azimuthal) oscillations within the inhomogeneous layer. We determine explicitly the decrement γ (decay time γ^-1) for a coronal flux tube whose plasma density varies only in a thin layer of thickness on the tube boundary. The effect of viscosity is also considered. We show that, in general, the problem involves two distinct timescales, γ^-1 and ωR^1/3, where R is the Reynolds number and ω_k is the frequency of the kink mode. Under coronal conditions (when γ^-1 ωR^1/3), the characteristic damping time of global oscillations is γ^-1. During this time, most of the energy in the initial perturbation is transferred into a resonant absorption layer of thickness of order ²/a, with motions in this layer having an amplitude of order a/ times the initial amplitude. We apply our results to the observations, suggesting that loop oscillations decay principally because of inhomogeneities in the loop. Our theory suggests that only those loops with density inhomogeneities on a small scale (confined to within a thin layer of order aγ/ω_k in thickness) are able to support coherent oscillations for any length of time and so be observable. Loops with a more gradual density variation, on the scale of the tube radius a, do not exhibit pronounced oscillations.