A principal has m identical objects to allocate among a group of n agents. Objects are desirable and each agent needs at most one copy. The principal's value of assigning an object to an agent is this agent's private information. There are no monetary transfers available but the principal can verify  up to k agents, where  k<m, upon which he perfectly learns the verified agent's type. The principal can penalize a lying agent by not assigning him an object. We find the mechanism that maximizes the principal's expected utility. In this mechanism, an agent gets an object if (i) his type is above a cutoff and among the m highest types, (ii) his type is above some lower cutoff but among the k highest types, or (iii) he gets an object in a lottery that allocates the remaining objects randomly. The verification rule can be chosen to make truthful reporting a Bayesian equilibrium.