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We propose a single-photon source based on a pair of weakly nonlinear optical cavities subject to a one-directional dissipative coupling. When both cavities are driven by mutually coherent fields, sub-Poissonian light is generated in the target cavity even when the nonlinear energy per photon is much smaller than the dissipation rate. The sub-Poissonian character of the field holds over a delay measured by the inverse photon lifetime, as in the conventional photon blockade, thus allowing single-photon emission under pulsed excitation. We discuss a possible implementation of the dissipative coupling relevant to photonic platforms.