  It is believed that millisecond pulsars attain their fast spins by accreting
matter and angular momentum from companion stars. Theoretical modelling of the
accretion process suggests a spin-up line in the period-period derivative
($P$-$\dot{P}$) diagram of millisecond pulsars, which plays an important role
in population studies of radio millisecond pulsars and accreting neutron stars
in X-ray binaries. Here we present observational evidence for such a spin-up
line using a sample of 143 radio pulsars with $P$ < 30 ms. We also find that
PSRs~J1823$-$3021A and J1824$-$2452A, located near the classic spin-up line,
are consistent with the broad population of millisecond pulsars. Finally, we
show that our approach of Bayesian inference can probe accretion physics,
allowing constraints to be placed on the accretion rate and the
disk-magnetosphere interaction.
