The Milky Way is known to be an abundant source of gamma-ray photons, now determined to be mainly diffuse in nature and resulting from interstellar processes. In the soft gamma-ray domain, point sources are expected to dominate, but the lack of sensitive high-resolution observations did not allow for a clear estimate of the contribution from such sources. Even the best imaging experiment revealed only a few point sources, accounting for about 50% of the total Galactic flux. Theoretical studies were unable to explain the remaining intense diffuse emission. Investigating the origin of the soft gamma-rays is therefore necessary to determine the dominant particle acceleration processes and to gain insights into the physical and chemical equilibrium of the interstellar medium. Here we report observations in the soft gamma-ray domain that reveal numerous compact sources. We show that these sources account for the entirety of the Milky Way's emission in soft gamma-rays, leaving at most a minor role for diffuse processes.