Highly sensitive phase- and frequency-resolved detection of microwave electric fields is of central importance in a wide range of fields, including cosmology(1,2), meteorology(3), communication(4) and microwave quantum technology(5). Atom-based electrometers(6,7) promise traceable standards for microwave electrometry, but their best sensitivity is currently limited to a few mu V cm(-1) Hz(-1/2) (refs. (8,9)) and they only yield information about the field amplitude and polarization(10). Here, we demonstrate a conceptually new microwave electric field sensor-the Rydberg-atom superheterodyne receiver (superhet). The sensitivity of this technique scales favourably, achieving even 55 nV cm(-1) Hz(-1/2) with a modest set-up. The minimum detectable field of 780 pV cm(-1) is three orders of magnitude smaller than what can be reached by existing atomic electrometers. The Rydberg-atom superhet allows SI-traceable measurements, reaching uncertainty levels of 10(-8) V cm(-1) when measuring a sub-mu V cm(-1) field, which has been inaccessible so far with atomic sensors. Our method also enables phase and frequency detection. In sensing Doppler frequencies, sub-mu Hz precision is reached for fields of a few hundred nV cm(-1). This work is a first step towards realizing electromagnetic-wave quantum sensors with quantum projection noise-limited sensitivity. Such a device will impact diverse areas like radio astronomy, radar technology and metrology. The Rydberg-atom superhet, based on microwave-dressed Rydberg atoms and a tailored electromagnetically induced transparency spectrum, allows SI-traceable measurements of microwave electric fields with unprecedented sensitivity.