  Motivated by the recent experiment of Velasco Jr. {\em et al.} [J. Velasco
Jr. {\em et al.}, Nat. Nanotechnology 7, {\bf 156} (2012)], we develop a
mean-field theory of the interaction-induced antiferromagnetic (AF) state in
bilayer graphene at charge neutrality point at arbitrary perpendicular magnetic
field B. We demonstrate that the AF state can persist at all $B$. At higher
$B$, the state continuously crosses over to the AF phase of the $\nu=0$ quantum
Hall ferromagnet, recently argued to be realized in the insulating $\nu=0$
state. The mean-field quasiparticle gap is finite at B=0 and grows with
increasing B, becoming quasi-linear in the quantum Hall regime, in accord with
the reported behavior of the transport gap. By adjusting the two free
parameters of the model, we obtain a simultaneous quantitative agreement
between the experimental and theoretical values of the key parameters of the
gap dependence -- its zero-field value and slope at higher fields. Our findings
suggest that the insulating state observed in bilayer graphene in Ref. 1 is
antiferromagnetic (canted, once the Zeeman effect is taken into account) at all
magnetic fields.
