The paramagnetic Meissner effect, previously observed in yttrium and bismuth cuprates, was discovered in DyRh3.8Ru0.2B4, a single-phase superconductor possessing its own intrinsic magnetic subsystem. Measurement of the magnetic moment in DyRh3.8Ru0.2B4 was carried out in the FC (field-cooling) and ZFC (zero-field cooling) modes: cooling in or without a magnetic field, followed by heating in the magnetic field. Appearance of a positive magnetic moment was established in the study of complex rhodium boride in the FC mode in the fields of ~20 Oe (below the first critical value). The appearance of a paramagnetic Meissner effect in DyRh3.8Ru0.2B4 at fields B < Bc1 is related to the peculiarities of the magnetic subsystem behavior when the sample is cooled in the FC or ZFC modes at a temperature below the phase transition point from the paramagnetic to the ferrimagnetic state. Correlation of this anomaly with the intrinsic magnetic subsystem of the compound formed by a cluster crystal structure of the LuRu4B4 type is discussed. The influence of the intrinsic magnetic subsystem on the increased stability of the superconducting state is established. The presence of the intrinsic magnetic subsystem distinguishes the studied single-phase rhodium boride from other type II superconductors, which do not have a magnetic subsystem.