In experiments and their interpretation usually the spin magnetic moment of magnons is considered. In this Letter, we identify a complementing orbital magnetic moment of magnons brought about by spin-orbit coupling. Our microscopic theory uncovers that spin magnetization M^{S} and orbital magnetization M^{O} are independent quantities; they are not necessarily collinear. Even when the total spin moment is compensated due to antiferromagnetism, M^{O} may be nonzero. This scenario of orbital weak ferromagnetism is realized in paradigmatic kagome antiferromagnets with Dzyaloshinskii-Moriya interaction. We demonstrate that magnets exhibiting a magnonic orbital moment are omnipresent and propose transport experiments for probing it.