This paper studies fully distributed parameter estimation under measurement attacks. A connected network of agents makes measurements of a parameter while an adversary manipulates a subset of the measurements. The goal of the agents is to recover the parameter in the presence of measurement attacks. This paper presents an iterative consen-sus+innovations algorithm for resilient distributed estimation. The algorithm ensures that all agents correctly recover the parameter of interest, with exponentially fast rate of convergence, so long as less than [3/10] of the agents' measurements are under attack, regardless of the (connected) network topology. We demonstrate the performance of the algorithm through numerical examples.