Altermagnets are magnetic materials with antiferromagnetic spin ordering but exhibit ferromagnetic

properties. Understanding the microscopic origin of the latter is a central problem. Ferromagnetlike

properties such as the anomalous Hall effect are linked with weak ferromagnetism, whose microscopic

origin in altermagnets remains unclear however. We show theoretically that the alternating g-tensor

anisotropy in altermagnets can induce weak orbital ferromagnetism even when the Dzyaloshinskii-Moriya

interaction is forbidden.We demonstrate this mechanism to explain weak ferromagnetism for both collinear

and noncollinear spin altermagnets. Our findings provide new insights into the origin of weak ferromagnetism

and suggest orbital-based ways for manipulating magnetic configurations in altermagnets.