The ground toaioncd Ievei structure of NH; in ammonium perrhenate (NIQRe04) has been analysed by a study of its proton magnetic resonance lineshape 51 the polycrystalhne state at 4 K. It is shown that, in this structure, tunneilin~ rotation of NH: around one of its C2 axes is greatly favocred over the others. 1. Introducrion The nuclear resonance properties of ammonium perrhenate (NH,ReO,) have attracted much attention in recent years [I -31. The structure of this compound at room temperature is known to belong to the tetragonal space group 141/a. Based on a study of the temperature dependence of the rhenium (Is7Re) quadrupole resonance transitions, Johnson and Rogers [ I] suggested a low-temperature (<240 K) phase transition in this compound which wt; related to the order-disorder nature of the NH:_ However, a subsequent detailed study [2] ofthe proton magnetic relaxation down to 60 K ruled out an order-disorder transition in this compound_ An additional significant result of the latter work, namely, that the ammonium ion was quite weakly hindered in this compound, led to the suggestion that quantum mechanical tunnelling rotations of NH: might be becoming important at temperatures of 60 K and below. Apart from the above results, a study of the details of the low temperature ammonium ion motions in this compound has not been made. in view of our interest in the nature of rotational tunnelhng motions of NH: in a variety of structures, we present here a lineshape