We study cross-lingual stance detection, which aims to leverage labeled data in one language to identify the relative perspective (or stance) of a given document with respect to a claim in a different target language. In particular, we introduce a novel contrastive language adaptation approach applied to memory networks, which ensures accurate alignment of stances in the source and target languages, and can effectively deal with the challenge of limited labeled data in the target language. The evaluation results on public benchmark datasets and comparison against current state-of-the-art approaches demonstrate the effectiveness of our approach.