  The cosmic proper distance $d_P$ is a fundamental distance in the Universe.
Unlike the luminosity and angular diameter distances, which correspond to the
angular size, the proper distance is the length of light path from the source
to observer. However, the proper distance has not been measured before. The
recent redshift measurement of a repeat fast radio burst (FRB) can shed light
on the proper distance. We show that the proper distance-redshift relation can
indeed be derived from dispersion measures (DMs) of FRBs with measured
redshifts. From Monte Carlo simulations, we find that about 500 FRBs with DM
and redshift measurements can tightly constrain the proper distance-redshift
relation. We also show that the curvature of our Universe can be constrained
with a model-independent method using this derived proper distance-redshift
relation and the observed angular diameter distances. Owing to the high event
rate of FRBs, hundreds of FRBs can be discovered in the future by upcoming
instruments. The proper distance will play an important role in investigating
the accelerating expansion and the geometry of the Universe.
