Integrated optical amplifiers are essential building blocks for emerging applications of large-scale and complex silicon photonic integrated circuits. Single-crystal erbium-incorporated rare-earth oxide thin films epitaxially grown on silicon substrate could be a promising material candidate. Based on the mechanism of bound states in the continuum, we have developed a low-loss waveguide platform for such materials. The waveguides are formed by patterning a silicon nitride cap layer grown on a rare-earth oxide thin film on silicon-on-insulator substrate. Low propagation loss of 4.70 dB/cm has been obtained in waveguides with silicon nitride/rare-earth oxide/silicon layer stack thickness of 72 nm/90 nm/300 nm, width of 1.08 µm, at a wavelength of 1445.8 nm. Optical signal enhancement in these waveguides upon optical pumping has been observed in a wide wavelength range of 1510 to 1560 nm, with a large peak enhancement of ∼16 dB/cm around 1536 nm. Our results represent an important step towards the realization of high-performance monolithic-integrated optical amplifiers on Si.