We report on the first complete system-level demonstration of a superconducting digital communication system. In today's digital computer and communication systems, managing data flow is a major challenge as the systems have to deal with a huge amount of information. A superconductor device may solve this problem. There have been several attempts to make superconducting switching core circuits, but there have been no complete system-level demonstrations. We have built such a system to demonstrate the efficiency of superconducting devices used in a communication system. Our system is focused on parallel processor communication, and consists of PCs as the processor elements at the three nodes, three interface boxes, and a superconductive chip immersed in a 4.2-K cryostat. The interconnection chip operation is essentially packet switching that was designed with a pipeline ring architecture. The interface box consists of parallel-serial converters, serial-parallel converters, a FIFO output buffer, and a level conversion circuit. The network system operates successfully at approximately 100 MHz, and the clock frequency is restricted by the speed of the interface ICs, not by that of the superconductive chip. We also confirmed the 2-GHz operation of the switching chip, and estimate that the total throughput of the system ran be increased to more than 10 Gbps.