  We theoretically investigate superconductivity under a spin-split band
structure owing to a collinear-type antiferromagnetic order in
quasi-two-dimensional organic compounds $\kappa$-(BEDT-TTF)$_2X$. We find that
the magnetic order can induce a Fulde--Ferrell--Larkin--Ovchinnikov (FFLO)
state, where the Cooper pair possesses a finite center-of-mass momentum. We
show this from two types of analyses: (1) an effective model where simple
intraband attractive interactions are assumed, and (2) many-body calculations
of the repulsive Hubbard model based on the fluctuation-exchange approximation
and the linearized Eliashberg equation. Our results show the possibility of
realizing the FFLO state without applying an external magnetic field.
