Ceramic-Al substrates with co-continuous ceramic and metal phases, which exhibit high thermal conductivity and compatible coefficient of thermal expansion (CTE), have been widely investigated through the process of die-casting. In this research, a kind of powder sintering process was proposed for fabricating ceramic-Cu composite substrates with co-continuous phases. Copper fiber (Cuf) has excellent thermal conductivity and large aspect ratio, making it an ideal material to form bridging network structures in the ceramic-Cu composite. To maintain the large aspect ratio of Cuf, and densify the composite substrate, ZnO-SiO2-CaO glass was introduced as a sintering additive. Both Al2O3/glass/Cuf and Al2O3/30glass/Cup composite substrates were hot-pressed at 850 °C under 25 MPa. Experimental results showed that the thermal conductivity of Al2O3/30glass/30Cuf composite substrate was as high as 38.9 W/mK, which was about 6 times that of Al2O3/30glass; in contrast, the thermal conductivity of Al2O3/30glass/30Cup composite substrate was only 25.9 W/mK. Microstructure observation showed that, influenced by hot press and corrosion of molten ZnO-SiO2-CaO glass, the copper fibers were deformed under hot-pressing, and some local melting-like phenomena occurred on the surface of copper fiber at 850 °C under 25 MPa. The molten phase originating from surface of Cuf welded the overlapping node of copper fibers during cooling process. Finally, the interconnecting metal bridging in ceramic matrix was formed and behaved as a rapid heat-dissipating channel, which is similar to substrates prepared through die-casting process by porous ceramic and melted Al.