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Phylogenetic systematics is an increasingly popular tool in historical linguistics for reconstructing the evolutionary histories of groups of languages. One problem in applying phylogenetic methods to languages is that phylogenetic methods assume evolution takes place strictly by descent with modification, whereas borrowing between languages is common. This paper tests two different methods for addressing borrowing in phylogenetic analysis of language on a dataset representing the dialects of ancient Greek: character weighting and preliminary cluster analysis. Both methods show promise; they correctly recovered the subgrouping of the Greek dialects and were able to improve the resolution of the tree compared to the preliminary analysis. However, they recovered conflicting subgroupings of the West Greek dialects. This result is most likely due to a circular dialect continuum within West Greek. Using phylogenetic methods in situations which match their assumptions is crucial; for the West Greek dialects, phylogenetic network methods would be more appropriate.