Quantum computers require classical electronics to ensure fault-tolerant operation. To address compactness and scalability, it was proposed to implement such electronics as integrated circuits operating at cryogenic temperatures close to those at which quantum bits (qubits) operate. Circulators are among the most common blocks used in the qubit readout chain, but they are currently discrete devices with a bulky footprint, thus preventing large-scale system integration. For this reason, we present here a detailed description of the first fully integrated CMOS circulator operating from 300 K down to 4.2 K to be an integral part of cryogenic quantum computing platforms. At 300 K, the circuit's operating frequency is centered around 6.5 GHz with 28% fractional bandwidth, and it has 2.2-dB insertion loss, 2.4-dB noise figure, and 18-dB isolation while consuming 2.5-mW core power. These results are achieved thanks to a fully passive architecture based on $LC$ all-pass filters, which allows achieving a $1.6\times $ increase in fractional bandwidth and the lowest power consumption with respect to the state of the art while using only 0.45 mm(2) of core area. This allows miniaturization of circulators in power-constrained multi-qubit readout systems.