In this paper, we introduce a robust algorithm \textsl{TranSync} for the 1D translation synchronization problem, which aims to recover the global coordinates of a set of nodes from noisy relative measurements along a pre-defined observation graph. The basic idea of TranSync is to apply truncated least squares, where the solution at each step is used to gradually prune out noisy measurements. We analyze TranSync under a deterministic noisy model, demonstrating its robustness and stability. Experimental results on synthetic and real datasets show that TranSync is superior to state-of-the-art convex formulations in terms of both efficiency an accuracy.