The growing availability of distributed data has led to the increased use of machine learning (ML) algorithms in distributed topologies, where multiple nodes collaborate to train models under the coordination of a central server. However, distributed learning faces two significant challenges: the risk of Byzantine nodes corrupting the learning process by sending incorrect information, and the potential for a curious server to violate the privacy of individual nodes, even reconstructing their private data. While homomorphic encryption (HE) has been a promising solution for privacy preservation in distributed settings, its high computational cost, especially for high-dimensional ML models, has made it challenging to design robust (non-linear) Byzantine-resilient algorithms using HE. In this paper, we introduce SABLE, the first distributed learning protocol that is both Byzantine robust and fully homomorphic. SABLE utilizes the novel Homomorphic Trimmed Sum (HTS) operator, which efficiently implements the robust coordinate-wise trimmed mean, providing strong defenses against Byzantine nodes while ensuring data privacy through HE. Extensive experiments on standard ML tasks show that SABLE achieves practical execution times and offers ML accuracy comparable to non-private approaches, proving its effectiveness in real-world scenarios.