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We present a complete structural analysis of the ellipticals (E), diffuse bulges (dB), compact bulges (cB), and disks (D) within a redshift range $0 < z < 1$, and stellar mass $\log_{10}(\mathrm{M}_*/\mathrm{M}_\odot) \geq 9.5$ volume-limited sample drawn from the combined DEVILS and HST-COSMOS region. We use the {\sc ProFit} code to profile over $\sim35,000$ galaxies for which visual classification into single or double-component was predefined in Paper-I. Over this redshift range, we see a growth in the total stellar mass density (SMD) of a factor of 1.5. At all epochs we find that the dominant structure, contributing to the total SMD, is the disk, and holds a fairly constant share of $\sim60\%$ of the total SMD from $z = 0.8$ to $z = 0.2$, dropping to $\sim30\%$ at $z = 0.0$ (representing $\sim33\%$ decline in the total disk SMD). Other classes (E, dB, and cB) show steady growth in their numbers and integrated stellar mass densities. By number, the most dramatic change across the full mass range is in the growth of diffuse bulges. In terms of total SMD, the biggest gain is an increase in massive elliptical systems, rising from 20\% at $z = 0.8$ to equal that of disks at $z = 0.0$ (30\%) representing an absolute mass growth of a factor of 2.5. Overall we see a clear picture of the emergence and growth of all three classes of spheroids over the past 8 Gyrs, and infer that in the later half of the Universe's timeline spheroid forming-processes and pathways (secular evolution, mass-accretion, and mergers) appear to dominate mass transformation over quiescent disk growth.