In single-crystal specimens of yttrium iron garnet doped with small amounts of ytterbium the ferrimagnetic resonance linewidths and the fields for resonance show anomalous behaviour in certain crystallographic directions at low temperatures. This paper presents some observations of this anomalous behaviour at 16·8 and 9·3 Gc/s as functions of temperature (1·5-25°K), concentration of ytterbium ions (1·, 5·1 and 10·2%) and direction of the applied magnetic field. The loci of the anomaly are deduced to be the (111) crystallographic planes, and some effects of thermal annealing are described. Some thoughts on the possible origin of the anomaly are also presented, from which it is found that many, though not all, features of the anomalous behaviour can be described by transverse resonance relaxation (with a relaxation time of approximately 5 × 10^-11 sec) occurring at a near-crossing of the low-lying energy levels (minimum energy separation about 0·8 cm^-1) of a small minority of Yb ions occupying `defect' sites.