High-frequency wave propagation problems arise in many applications of science and engineering. This paper investigates the adaptive numerical solution of the Helmholtz equation at high wavenumber by a particular discontinuous Galerkin (dG) method based on the ultra weak formulation of O. Cessenat and B. Deprés. An important feature of this discretization is its unconditional stability with respect to the wave number. \par The main focus of the paper lies on the construction and analysis of a~posteriori error estimates for the dG method. Particular emphasis is put on deriving estimates that are explicit with respect to the meshsize $h$, the polynomial degree $p$, and the wavenumber $k$. The proposed estimates do not require a strict resolution condition and the a~posteriori estimator is thus applicable on very coarse meshes and allows adaptive refinement starting from very coarse initial meshes. \par The robust performance of the proposed methods is illustrated with numerical results for a couple of test problems, including problems with singularities and with varying wavenumbers.