Superconducting qubits 1, 2 are one of the most promising candidates to implement quantum computers, with projected applications in physics simulations 3 , optimization 4 , machine learning 5 , and chemistry 6 . The superiority of superconducting quantum computers over any classical device in simulating random but well-determined quantum circuits has already been shown in two independent experiments 7, 8 and important steps have been taken in quantum error correction 9-11 . However, the currently wide-spread qubit designs [12][13][14][15][16] do not yet provide high enough performance to enable practical applications or efficient scaling of logical qubits owing to one or several following issues: sensitivity to charge or flux noise leading to decoherence, too weak non-linearity preventing fast operations, undesirably dense excitation spectrum, or complicated design vulnerable to parasitic capacitance. Here, we introduce and demonstrate a superconducting-qubit type, the unimon, which combines the desired properties of high non-linearity, full insensitivity to dc charge noise, insensitivity to flux noise, and a simple structure consisting only of a single Josephson junction in a resonator. We measure the qubit frequency, ω 01 /(2π), and anharmonicity α over the full dc-flux range and observe, in agreement with our quantum models, that the qubit anharmonicity is greatly enhanced at the optimal operation point, yielding, for example, 99.9% and 99.8% fidelity for 13-ns single-qubit gates on two qubits with (ω 01 , α) = (4.49 GHz, 434 MHz) × 2π and (3.55 GHz, 744 MHz) × 2π, respectively. The energy relaxation time T 1 10 µs is stable for hours and seems to be limited by dielectric losses. Thus, future improvements of the design, materials, and gate time may promote the unimon to break the 99.99% fidelity target for efficient quantum error correction and possible quantum advantage with noisy systems.