We report the discovery of a large population of ultra-diffuse galaxies (UDGs) in the massive galaxy cluster Abell 2744 (z = 0.308) as observed by the Hubble Frontier Fields program. Since this cluster is ~5 times more massive than Coma, our observations allow us to extend 0.7 dex beyond the high-mass end of the relationship between UDG abundance and cluster mass reported by van der Burg et al. Using the same selection criteria as van der Burg et al., A2744 hosts an estimated 1961 ± 577 UDGs, 10 times the number in Coma. As noted by Lee & Jang, A2744 contains numerous unresolved compact objects, which those authors identified predominantly as globular clusters. However, these objects have luminosities that are more consistent with ultra-compact dwarf (UCD) galaxies. The abundances of both UCDs and UDGs scale with cluster mass as a power law with a similar exponent, although UDGs and UCDs have very different radial distributions within the cluster. The radial surface density distribution of UCDs rises sharply toward the cluster center, while the surface density distribution of the UDG population is essentially flat. Together, these observations hint at a picture where some UCDs in A2744 may have once been associated with infalling UDGs. As UDGs fall in and dissolve, they leave behind a residue of unbound UCDs.