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Although biometric facial recognition systems are fast becoming part of security applications, these systems are still vulnerable to morphing attacks, in which a facial reference image can be verified as two or more separate identities. In border control scenarios, a successful morphing attack allows two or more people to use the same passport to cross borders. In this paper, we propose a novel differential morph attack detection framework using a deep Siamese network. To the best of our knowledge, this is the first research work that makes use of a Siamese network architecture for morph attack detection. We compare our model with other classical and deep learning models using two distinct morph datasets, VISAPP17 and MorGAN. We explore the embedding space generated by the contrastive loss using three decision making frameworks using Euclidean distance, feature difference and a support vector machine classifier, and feature concatenation and a support vector machine classifier.