  We present the first measurement of the gravitational quadrupole moment of
the companion star of a spider pulsar, namely the black widow PSR J2051$-$0827.
To this end we have re-analysed radio timing data using a new model which is
able to account for periastron precession caused by tidal and centrifugal
deformations of the star as well as by general relativity. The model allows for
a time-varying component of the quadrupole moment, thus self-consistently
accounting for the ill-understood orbital period variations observed in these
systems. Our analysis results in the first detection of orbital precession in a
spider system at $\dot{\omega} = -68.6_{-0.5}^{+0.9}$ deg/yr and the most
accurate determination of orbital eccentricity for PSR J2051$-$0827 with $e =
(4.2 \pm 0.1) \times 10^{-5}$. We show that the variable quadrupole component
is about 100 times smaller than the average quadrupole moment $\bar{Q} =
-2.2_{-1}^{+0.6} \times 10^{41}$ kg.m$^2$. We discuss how accurate modelling of
high precision optical light curves of the companion star will allow its
apsidal motion constant to be derived from our results.
