  Gravitational-wave cosmology began in 2017 with the observation of the
gravitational waves emitted in the merger of two neutron stars, and the
coincident observation of the electromagnetic emission that followed. Although
only a $30\%$ measurement of the Hubble constant was achieved, future
observations may yield more precise measurements either through other
coincident events or through cross correlation of gravitational-wave events
with galaxy catalogs. Here, we implement a new way to measure the Hubble
constant without an electromagnetic counterpart and through the use of the
binary Love relations. These relations govern the tidal deformabilities of
neutron stars in an equation-of-state insensitive way. Importantly, the Love
relations depend on the component masses of the binary in the source frame.
Since the gravitational-wave phase and amplitude depend on the chirp mass in
the observer (and hence redshifted) frame, one can in principle combine the
binary Love relations with the gravitational-wave data to directly measure the
redshift, and thereby infer the value of the Hubble constant. We implement this
approach in both real and synthetic data through a Bayesian parameter
estimation study in a range of observing scenarios. We find that for the
LIGO/Virgo/KAGRA design sensitivity era, this method results in a similar
measurement accuracy of the Hubble constant to those of current-day, dark-siren
measurements. For third generation detectors, this accuracy improves to
$\lesssim 10\%$ when combining measurements from binary neutron star events in
the LIGO Voyager era, and to $\lesssim 2\%$ in the Cosmic Explorer era.
