A new distributed asynchronous offset correction algorithm for time synchronization in networks with random communication delays, communication dropouts, and measurement noise is proposed. The algorithm is based on a specially constructed error function which includes delay compensation and unboundedly-increasing-time compensation parameters. Convergence of the algorithm in the mean square sense and w.p.l is proved. A modification of the algorithm based on consensus on the delay compensation parameter is also proposed, offering better performance in practice. Illustrative simulation results are presented.