Consider the problem of estimating the location vector and scatter matrix from a set of multivariate data. Two standard classes of robust estimates are M-estimates and S-estimates. The M-estimates can be tuned to give good local robustness properties, such as good efficiency and a good bound on the influence function at an underlying distribution such as the multivariate normal. However, M-estimates suffer from poor breakdown properties in high dimensions. On the other hand, S-estimates can be tuned to have good breakdown properties, but when tuned in this way, they tend to suffer from poor local robustness properties. In this paper a hybrid estimate called a constrained M-estimate is proposed which combines both good local and good global robustness properties.