We follow the structural evolution of star forming galaxies (SFGs) like the Milky Way by selecting progenitors to z~1.3 based on the stellar mass growth inferred from the evolution of the star forming sequence. We select our sample from the 3D-HST survey, which utilizes spectroscopy from the HST WFC3 G141 near-IR grism and enables precise redshift measurements for our sample of SFGs. Structural properties are obtained from Sersic profile fits to CANDELS WFC3 imaging. The progenitors of z=0 SFGs with stellar mass M=10^{10.5} Msun are typically half as massive at z~1. This late-time stellar mass assembly is consistent with recent studies that employ abundance matching techniques. The descendant SFGs at z~0 have grown in half-light radius by a factor of ~1.4 since z~1. The half-light radius grows with stellar mass as r_e M^{0.29}. While most of the stellar mass is clearly assembling at large radii, the mass surface density profiles reveal ongoing mass growth also in the central regions where bulges and pseudobulges are common features in present day late-type galaxies. Some portion of this growth in the central regions is due to star formation as recent observations of H-alpha maps for SFGs at z~1 are found to be extended but centrally peaked. Connecting our lookback study with galactic archeology, we find the stellar mass surface density at R=8 kpc to have increased by a factor of ~2 since z~1, in good agreement with measurements derived for the solar neighborhood of the Milky Way.
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