Video Anomaly Detection (VAD) automates the identification of unusual events, such as security threats in surveillance videos. In real-world applications, VAD models must effectively operate in cross-domain settings, identifying rare anomalies and scenarios not well-represented in the training data. However, existing cross-domain VAD methods focus on unsupervised learning, resulting in performance that falls short of real-world expectations. Since acquiring weak supervision for the source domain is cost-effective, we conjecture that combining it with external unlabeled data has notable potential to enhance cross-domain performance. To this end, we introduce a novel weakly supervised framework for Cross-Domain Learning (CDL) in VAD that incorporates external data during training by estimating its prediction bias and adaptively minimizing that using the predicted uncertainty. We demonstrate the effectiveness of the proposed CDL framework through comprehensive experiments conducted in various configurations on two large-scale VAD datasets: UCF-Crime and XD-Violence. Our method significantly surpasses the state-of-the-art works in cross-domain evaluations, achieving an average absolute improvement of 19.6% on UCF-Crime and 12.87% on XD-Violence.