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Quantum key distribution (QKD) enables the establishment of secret keys between users connected via a channel vulnerable to eavesdropping, with information-theoretic security, that is, independently of the power of a malevolent party. QKD systems based on the encoding of the key information on continuous variables (CV), such as the values of the quadrature components of coherent states, present the major advantage that they only require standard telecommunication technology. However, the most general security proofs for CV-QKD required until now the use of Gaussian modulation by the transmitter, complicating practical implementations. Here, we experimentally implement a protocol that allows for arbitrary, Gaussian-like, discrete modulations, whose security is based on a theoretical proof that applies very generally to such situations. These modulation formats are compatible with the use of powerful tools of coherent optical telecommunication, allowing our system to reach a performance of tens of megabit per second secret key rates over 25 km.