As digital services are increasingly being deployed and used in a variety of domains, the environmental impact of Information and Communication Technologies (ICTs) is a matter of concern. Artificial intelligence is driving some of this growth but its environmental cost remains scarcely studied. A recent trend in large-scale generative models such as ChatGPT has especially drawn attention since their training requires intensive use of a massive number of specialized computing resources. Generative AI (Gen-AI) represents a new stage in digital transformation through its many applications. Unfortunately, by accelerating the growth of digital technology, Gen-AI is contributing to the multiple environmental damages caused by its sector. The question of the sustainability of IT must include this new technology and its applications, by estimating its environmental impact. This work proposes a methodology for a multi-criteria evaluation of the environmental impact of generative AI services, considering embodied and usage costs of all the resources required for training models, inferring from them, and hosting them online. Combining life-cycle analysis (LCA) methods and direct measurement experiments, we illustrate our methods by studying Stable Diffusion, an open-source text-to-image Gen-AI model accessible online as a service. This use case is based on an experimental observation of training and inference energy consumption of the model. By calculating the full environmental costs of this Gen-AI service from end to end, we broaden our view of the impact of these technologies. We show that Gen-AI, as a service, generates an impact through the use of numerous user terminals and networks. We also show that decarbonizing the sources of electricity for these services will not be enough to solve the problem of their sustainability, due to their consumption of energy and rare metals. This consumption will inevitably raise the question of feasibility in a world of finite resources. Various scenarios estimating the influence of usage intensity on the impact sources are explored through a sensitivity analysis.