We present a device demonstrating a lithographically patterned transmon integrated with a micromachined cavity resonator. Our two-cavity, one-qubit device is a multilayer microwave-integrated quantum circuit (MMIQC), comprising a basic unit capable of performing circuit-QED operations. We describe the qubit-cavity coupling mechanism of a specialized geometry using an electric-field picture and a circuit model, and obtain specific system parameters using simulations. Fabrication of the MMIQC includes lithography, etching, and metallic bonding of silicon wafers. Superconducting wafer bonding is a critical capability that is demonstrated by a micromachined storage-cavity lifetime of 34.3  𝜇⁢s, corresponding to a quality factor of 2×106 at single-photon energies. The transmon coherence times are 𝑇₁=6.4  𝜇⁢s, and 𝑇₂^echo=11.7  𝜇⁢s. We measure qubit-cavity dispersive coupling with a rate 𝜒_𝑞⁢𝜇/2⁢𝜋=−1.17  MHz, constituting a Jaynes-Cummings system with an interaction strength 𝑔/2⁢𝜋=49  MHz. With these parameters we are able to demonstrate circuit-QED operations in the strong dispersive regime with ease. Finally, we highlight several improvements and anticipated extensions of the technology to complex MMIQCs.