From telecommunication to computing architectures, the realm of classical information hinges on converter technology to enable the exchange of data between digital and analog formats, a process now routinely performed across a variety of electronic devices. A similar exigency exists as well in quantum information technology where different frameworks are being developed for quantum computing, communication, and sensing. Thus, efficient quantum interconnects are a major need to bring these parallel approaches together and scale up quantum information systems. So far, however, the conversion between different optical quantum-bit encodings has remained challenging due to the difficulty of preserving fragile quantum superpositions and the demanding requirements for postselection-free implementations. Here we demonstrate such a conversion of quantum information between the two main paradigms, namely discrete- and continuous-variable qubits. We certify the protocol on a complete set of single-photon qubits, successfully converting them to cat-state qubits with fidelities exceeding the classical limit. Our result demonstrates an essential tool for enabling interconnected quantum devices and architectures with enhanced versatility and scalability.