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Splenomegaly, the enlargement of the spleen, is an important clinical indicator for various associated medical conditions, such as sickle cell disease (SCD). Spleen length measured from 2D ultrasound is the most widely used metric for characterising spleen size. However, it is still considered a surrogate measure, and spleen volume remains the gold standard for assessing spleen size. Accurate spleen volume measurement typically requires 3D imaging modalities, such as computed tomography or magnetic resonance imaging, but these are not widely available, especially in the Global South which has a high prevalence of SCD. In this work, we introduce a deep learning pipeline, DeepSPV, for precise spleen volume estimation from single or dual 2D ultrasound images. The pipeline involves a segmentation network and a variational autoencoder for learning low-dimensional representations from the estimated segmentations. We investigate three approaches for spleen volume estimation and our best model achieves 86.62%/92.5% mean relative volume accuracy (MRVA) under single-view/dual-view settings, surpassing the performance of human experts. In addition, the pipeline can provide confidence intervals for the volume estimates as well as offering benefits in terms of interpretability, which further support clinicians in decision-making when identifying splenomegaly. We evaluate the full pipeline using a highly realistic synthetic dataset generated by a diffusion model, achieving an overall MRVA of 83.0% from a single 2D ultrasound image. Our proposed DeepSPV is the first work to use deep learning to estimate 3D spleen volume from 2D ultrasound images and can be seamlessly integrated into the current clinical workflow for spleen assessment.