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Superconducting qubits are one of the most promising candidates to implement quantum computers. 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 and important steps have been taken in quantum error correction. However, the currently wide-spread qubit designs 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, $\omega_{01}/(2\pi)$, and anharmonicity $\alpha$ 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 $(\omega_{01},\alpha)=(4.49~\mathrm{GHz}, 434~\mathrm{ MHz})\times 2\pi$ and $(3.55~\mathrm{GHz}, 744~\mathrm{ MHz})\times 2\pi$, respectively. The energy relaxation time $T_1\lesssim 10~\mu\mathrm{s}$ 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., Comment: Main text: 37 pages,10 figures, 3 tables. Supplementary: 34 pages, 8 figure, 1 table