With muon spin rotation (µSR) we studied the transition between the orthorhombic antiferromagnetic (o-AF) and the tetragonal antiferromagnetic (t-AF) states of a weakly underdoped Ba1−xKxFe2As2 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-Q magnetic structure with c-axis oriented moments which has a vanishing magnetic moment on half of the Fe sites.The phase diagram of the iron arsenide superconductors is characterized by the close proximity of the antiferromagnetic (AF) and superconducting (SC) orders [1][2][3][4][5]. In the underdoped regime they coexist and compete for the same low-energy electronic states [6,7]. Moreover, the SC transition temperature, T c , reaches its maximum near the point where the AF order vanishes. It is therefore commonly assumed that the AF fluctuations are involved in the SC pairing mechanism [4,5]. Accordingly, it is important to obtain a full understanding of the AF ground state and also of the nearly degenerate states which can strongly impact the magnetic fluctuations.For the 122-type BaFe 2 As 2 system, the primary AF order is a so-called single-Q state with in-plane antiparallel spins along the (0, π) direction and parallel ones along (π, 0) [1][2][3]8]. The transition to this so-called "stripelike" AF state is coupled with a structural transition from a tetragonal paramagnetic to an orthorhombic AF (o-AF) state. The interactions underlying this magnetostructural transition are the subject of an ongoing discussion. The explanations range from the so-called itinerant models for which the Fermi-surface nesting governs the magnetic interactions [4,5,9] to orbital models which assume localized spins whose exchange interactions are determined by the orbital occupation [10,11]. An important role of the spin-lattice coupling [12] and a near degeneracy of different spin states of the Fe ions have also been discussed [13,14].In view of this ongoing discussion, the recent observation of a phase transition from the o-AF state to a new type of AF state with tetragonal lattice symmetry (t-AF) has attracted considerable attention. This t-AF state was first observed at ambient pressure in Ba 1−x Na x Fe 2 As 2 (BNFA) [15,16] and more recently in Ba 1−x K x Fe 2 As 2 (BKFA) [17,18] in a narrow doping range close to optimal doping. A state with very similar phenomenology was observed earlier in BKFA at elevated pressures [19]. In BKFA, even a reentrance of the o-AF state at lower temperature has been observed which seems to be induced (or at least supported) by superconductivity [17,20].Motivated by these observations, a range of so-called double-Q magnetic structures have been proposed for the t-AF state [21][22][23]. These correspond to different superpositions of the single-Q states with wave vectors (0, π) and (π, 0). A perpendicular magnetis...