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A new variant of SARS-CoV-2 (VOC-202012/01, here called VOC) emerged in late 2020 in the United Kingdom. Preliminary estimates indicate that 1.4% of newly diagnosed COVID-19 cases in w01 in the country are due to VOC. Using the model developed by INSERM to respond to the COVID-19 pandemic, we provide projections of the epidemic trajectories in the upcoming weeks due to increased VOC circulation in the country. We estimate that the variant would become dominant in France between late February and mid-March, depending on the epidemic evolution and estimated increase of VOC transmissibility. New weekly hospitalizations are expected to reach the level of the first wave’s peak (approximately 25,000 hospitalizations) between mid-February and early April, in absence of interventions. These results support the need for strengthened social distancing measures and for accelerating the vaccination campaign to face the threat of the VOC variant.