Neural Weather Models (NWM) are novel data-driven weather forecasting tools based on neural networks that have recently achieved comparable deterministic forecast skill to current operational approaches using significantly less real-time computational resources. The short inference times required by NWMs allow the generation of a large ensemble potentially providing benefits in quantifying the forecast uncertainty, particularly for extreme events, which is of critical importance for various socio-economic sectors. Here we propose a novel ensemble design for NWMs spanning two main sources of uncertainty: epistemic —or model uncertainty,— and aleatoric —or initial condition uncertainty. For the epistemic uncertainty, we propose an effective strategy for creating a diverse ensemble of NWMs that captures uncertainty in key model parameters. For the aleatoric, we explore the “breeding of growing modes” for the first time on NWMs, a technique traditionally used for operational numerical weather predictions as an estimate of the initial condition uncertainty. The combination of these two types of uncertainty produces an ensemble of NWM-based forecasts that is shown to improve upon benchmark probabilistic NWM and is competitive with the 51-member ensemble of the European Centre for Medium-Range Weather Forecasts based on the Integrated Forecasting System (IFS) —a gold standard in weather forecasting,— in terms of both error and calibration. In addition, we report better probabilistic skill than the IFS over land for two key variables: surface wind and air surface temperature.