A class of stationary, rigidly rotating perfect fluids coupled with nonlinear electromagnetic fields was investigated. An exact solution of the Einstein equations with sources for the Carter B(+) branch was found for the equation of state 3p + ¢ = const. We use a structure function for the Born-lnfeld nonlinear electrodynamics which is invariant under duality rotations and a metric possessing a four-parameter group of motions. The solution is of Petrov type D and the eigenvectors of the electromagnetic field are aligned to the Debever-Penrose vectors. 1259 0020-7748/96/0600-t259509.5010 © 1996 Plenum Publishing Coq~oralion m = b2f(X), X ----~ (D z + B2), b = const