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We design and test a low-loss interface between superconducting three-dimensional microwave cavities and two-dimensional circuits, where the coupling rate is highly tunable. This interface seamlessly integrates a loop antenna and a Josephson junction-based coupling element. We demonstrate that the loss added by connecting this interface to the cavity is 1.28 kHz, corresponding to an inverse quality factor of 1/(4.5×106). Furthermore, we show that the cavity's external coupling rate to a 50 Ω transmission line can be tuned from negligibly small to over 3 orders of magnitude larger than its internal loss rate in a characteristic time of 3.2 ns. This switching speed does not impose additional limits on the coupling rate because it is much faster than the coupling rate. Moreover, the coupler can be controlled by low frequency signals to avoid interference with microwave signals near the cavity or qubit frequencies. Finally, the coupling element introduces a 0.04 Hz/photon self-Kerr nonlinearity to the cavity, remaining linear in high photon number operations.