The low uptake of low-carbon heating systems across Europe has prompted authorities to consider more ambitious measures, including a complete ban on the installation of new fossil fuel boilers. In this analysis, we simulate the impacts of introducing a ban on gas boilers under 11,664 scenarios covering major uncertainties. Taking France as a case study, we demonstrate that a ban is critical to meet carbon neutrality. The ban achieves a reduction in total system cost in over 75% of scenarios, as costly investments in heat pumps are offset by a more efficient energy system. We demonstrate that the implementation of the ban, when coupled with the existing subsidy frame- work, mitigates inequalities among owner-occupied households but generate adverse affect for those in privately rented homes. Overall, our findings support the introduction of a ban on gas boilers, emphasizing the need to assess policies not only for their cost-effectiveness but also for their robustness to address uncertainties.