  Galaxy mergers produce pairs of supermassive black holes (SMBHs), which may
be witnessed as dual quasars if both SMBHs are rapidly accreting. The
kiloparsec (kpc)-scale separation represents a physical regime sufficiently
close for merger-induced effects to be important yet wide enough to be directly
resolvable with the facilities currently available. Whereas many kpc-scale dual
active galactic nuclei--the low-luminosity counterparts of quasars--have been
observed in low-redshift mergers, no unambiguous dual quasar is known at cosmic
noon ($z\approx2$), the peak of global star formation and quasar activity. Here
we report multiwavelength observations of SDSS J0749+2255 as a kpc-scale,
dual-quasar system hosted by a galaxy merger at cosmic noon ($z=2.17$). We
discover extended host galaxies associated with the much brighter compact
quasar nuclei (separated by 0.46" or 3.8 kpc) and low-surface-brightness tidal
features as evidence for galactic interactions. Unlike its low-redshift and
low-luminosity counterparts, SDSS J0749+2255 is hosted by massive compact
disk-dominated galaxies. The apparent lack of stellar bulges and the fact that
SDSS J0749+2255 already follows the local SMBH mass--host stellar mass
relation, suggest that at least some SMBHs may have formed before their host
stellar bulges. While still at kpc-scale separations where the host-galaxy
gravitational potential dominates, the two SMBHs may evolve into a
gravitationally bound binary system in around 0.22 Gyr.
