  We prove that the predual of any von Neumann algebra is $1$-Plichko, i.e., it
has a countably $1$-norming Markushevich basis. This answers a question of the
third author who proved the same for preduals of semifinite von Neumann
algebras. As a corollary we obtain an easier proof of a result of U.~Haagerup
that the predual of any von Neumann algebra enjoys the separable
complementation property. We further prove that the self-adjoint part of the
predual is $1$-Plichko as well.
