Morphed face images have recently become a growing concern for existing face verification systems, as they are relatively easy to generate and can be used to impersonate someone's identity for various malicious purposes. Efficient Morphing Attack Detection (MAD) that generalizes well across different morphing techniques is, therefore, of paramount importance. Existing MAD techniques predominantly rely on discriminative models that learn from examples of bona fide and morphed images and, as a result, often exhibit sub-optimal generalization performance when confronted with unknown types of morphing attacks. To address this problem, we propose a novel, diffusion-based MAD method in this paper that learns only from the characteristics of bona fide images. Various forms of morphing attacks are then detected by our model as out-of-distribution samples. We perform rigorous experiments over four different datasets (CASIA-WebFace, FRLL-Morphs, FERET-Morphs and FRGC-Morphs) and compare the proposed solution to both discriminatively-trained and once-class MAD models. The experimental results show that our MAD model achieves highly competitive results on all considered datasets.