A new mechanism for electromagnetic emission in the terahertz (THz) frequency regime from laser-plasma interactions is described. A localized and long-lasting transverse current is produced by two counter-propagating short laser pulses in weakly magnetized plasma. We show that the electromagnetic wave radiating from this current source, even though its frequency is close to cut-off of the ambient plasma, grows and diffuses towards the plasma-vacuum boundary, emitting a strong monochromatic THz wave. With driving laser pulses of moderate power, the THz wave has a field strength of tens of MV m−1, a frequency of a few THz and a quasi-continuous power that exceeds all previous monochromatic THz sources. The novelty of the mechanism lies in a diffusing electromagnetic wave close to cut-off, which is modelled by a continuously driven complex diffusion equation.