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We present detailed submillimeter- through centimeter-wave observations of the extraordinary extragalactic transient AT2018cow. The apparent characteristics—the high radio luminosity, the rise and long-lived emission plateau at millimeter bands, and the sub-relativistic velocity—have no precedent. A basic interpretation of the data suggests E(k) ≳ 4 X 10^(48) erg coupled to a fast but sub-relativistic (v ≈ 0.13c) shock in a dense (n(e) ≈ 3 X 10^(5) per cu. cm) medium. We find that the X-ray emission is not naturally explained by an extension of the radio-submm synchrotron spectrum, nor by inverse Compton scattering of the dominant blackbody UV/optical/IR photons by energetic electrons within the forward shock. By ∆t ≈ 20 days, the X-ray emission shows spectral softening and erratic inter-day variability. Taken together, we are led to invoke an additional source of X-ray emission: the central engine of the event. Regardless of the nature of this central engine, this source heralds a new class of energetic transients shocking a dense medium, which at early times are most readily observed at millimeter wavelengths