The Matrix Element Method has proven to be a powerful method to optimally exploit the information available in detector data. Its widespread use is nevertheless impeded by its complexity and the associated computing time. MoMEMta, a C++ software package to compute the integrals at the core of the method, provides a versatile implementation of the Matrix Element Method to both the theory and experiment communities. Its modular structure covers the needs of experimental analysis workflows at the LHC without compromising ease of use on simpler and smaller simulated samples used for phenomenological studies. With respect to existing tools, MoMEMta improves on usability and flexibility. In this paper, we present version 1.0 of MoMEMta, together with examples illustrating the wide range of applications at the LHC accessible for the first time with a single tool.