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To date, the highest fidelity quantum logic gates between two qubits have been achieved with variations on the geometric-phase gate in trapped ions, with the two leading variants being the Molmer-Sorensen gate and the light-shift (LS) gate. Both of these approaches have their respective advantages and challenges. For example, the latter is technically simpler and is natively insensitive to optical phases, but it has not been made to work directly on a clock-state qubit. We present a new technique for implementing the LS gate that combines the best features of these two approaches: By using a small ($\sim {\rm MHz}$) detuning from a narrow (dipole-forbidden) optical transition, we are able to operate an LS gate directly on hyperfine clock states, achieving gate fidelities of $99.74(4)\%$ using modest laser power at visible wavelengths. Current gate infidelities appear to be dominated by technical noise, and theoretical modeling suggests a path towards gate fidelity above $99.99\%$.