This paper deals with the problem of estimating the location parameter of a two parameter exponential distribution in case of contaminated data. Since in this case the sample minimum is an extremely unreliable estimator, robust alternatives are necessary. We investigate two types of estimators closer. The first type is based on a simple relation for the median. The second type has originally been suggested for Type-II-censored samples, but it also has good robustness properties. We discuss the breakdown properties of the two types of estimators and compare their performance for various patterns of data contamination in an extensive simulation study.