  With muon spin rotation ($ \mu $SR) we studied the transition between the
orthorhombic antiferromagnetic (o-AF) and the tetragonal antiferromagnetic
(t-AF) states of a weakly underdoped Ba$ _{1-x} $K$ _{x} $Fe$ _{2} $As$ _{2} $
single crystal. We observed some characteristic changes of the magnitude and
the orientation of the magnetic field at the muon site which, due to the fairly
high point symmetry of the latter, allow us to identify the magnetic structure
of the t-AF state. It is the so-called, inhomogeneous double-$\mathbf{Q}$
magnetic structure with $ c $-axis oriented moments which has a vanishing
magnetic moment on half of the Fe sites.
