Numerical simulations give evidence that a weak microundulator (field on axis: Bu≊50 G, period: λu≊5 mm), operating in magnetoresonance with a strong guide field (Bg≊3×104 G), is capable of producing millimeter and submillimeter radiation with an output power around 1 MW. The electronic efficiency appears to range between 1% and 5%, depending on the radial profile of the electron beam density at the undulator entrance. As an example, the energy of the electron beam is 300 kV to yield a radiation frequency slightly below 200 GHz. Analytical scaling laws for the electron dynamics at magnetoresonance predict that the relative spectral width of the spontaneous emission is proportional to (Bu/Bg)2/3, thus giving support to the surprising results in the case of a weak microundulator.