  Let $x$ and $y$ be two unit vectors in a normed plane $\mathbb{R}^2$. We say
that $x$ is Birkhoff orthogonal to $y$ if the line through $x$ in the direction
$y$ supports the unit disc. A B-measure (Fankh\"anel 2011) is an angular
measure $\mu$ on the unit circle for which $\mu(C)=\pi/2$ whenever $C$ is a
shorter arc of the unit circle connecting two Birkhoff orthogonal points. We
present a characterization of the normed planes that admit a B-measure.
