  Galaxies with the mass of the Milky Way dominate the stellar mass density of
the Universe but it is uncertain how and when they were assembled. Here we
study progenitors of these galaxies out to z=2.5, using data from the 3D-HST
and CANDELS Treasury surveys. We find that galaxies with present-day stellar
masses of log(M)~10.7 built ~90% of their stellar mass since z=2.5, with most
of the star formation occurring before z=1. In marked contrast to the assembly
history of massive elliptical galaxies, mass growth is not limited to large
radii: the mass in the central 2 kpc of the galaxies increased by a factor of
3.2+-0.8 between z=2.5 and z=1. We therefore rule out simple models in which
bulges were fully assembled at high redshift and disks gradually formed around
them. Instead, bulges (and black holes) likely formed in lockstep with disks,
through bar instabilities, migration, or other processes. We find that after
z=1 the growth in the central regions gradually stopped and the disk continued
to build, consistent with recent studies of the gas distributions in z~1
galaxies and the properties of many spiral galaxies today.
