In this letter, the time-difference-of-arrival-based localization problem under mixed line-of-sight (LOS)/non-LOS (NLOS) conditions is addressed. Under the assumption that the path status information is known, we formulate a robust weighted-least-squares method to solve this problem. To fully utilize the more accurate LOS measurements, we impose a weight to the term with respect to the NLOS measurements, and the choice of the weight is derived and explicitly given. We then employ the second-order cone relaxation technique to relax the problem as a tractable second-order cone program. Simulation results show that the prior path status information significantly improves the localization performance.