  The superconducting fluxonium qubit has a great potential for high-fidelity
quantum gates with its long coherence times and strong anharmonicity at the
half flux quantum sweet spot. However, current implementations of two-qubit
gates compromise fluxonium's coherence properties by requiring either a
temporary population of the non-computational states or tuning the magnetic
flux off the sweet spot. Here we realize a fast all-microwave cross-resonance
gate between two capacitively-coupled fluxoniums with the qubit dynamics well
confined to the computational space. We demonstrate a direct CNOT gate in 70 ns
with fidelity up to $\mathcal{F}=0.9949(6)$ despite the limitations of a
sub-optimal measurement setup and device coherence. Our results project a
possible pathway towards reducing the two-qubit error rate below $10^{-4}$ with
present-day technologies.
