  We use the full-mission Planck PR4 data to measure the CMB lensing
convergence ($\kappa$)--thermal Sunyaev-Zel'dovich (tSZ, $y$)
cross-correlation, $C_\ell^{y\kappa}$. This is only the second measurement to
date of this signal, following Hill \& Spergel (2014). We perform the
measurement using foreground-cleaned tSZ maps built from the PR4 frequency maps
via a tailored needlet internal linear combination (NILC) code in our companion
paper, in combination with the Planck PR4 $\kappa$ maps and various
systematic-mitigated PR3 $\kappa$ maps. A serious systematic is the residual
cosmic infrared background (CIB) in the tSZ map, as the high CIB--$\kappa$
correlation can significantly bias the inferred tSZ--$\kappa$
cross-correlation. We mitigate this by deprojecting the CIB in our NILC, using
a moment-deprojection approach to avoid leakage due to incorrect modelling of
the CIB frequency dependence. We validate our method on mm-sky simulations. We
fit a theoretical halo model to our measurement, finding a best-fit amplitude
of $A=0.82\pm0.21$ (for the highest signal-to-noise PR4 $\kappa$ map) or
$A=0.56\pm0.24$ (for a PR3 $\kappa$ map built from a tSZ-deprojected CMB map),
indicating that the data are consistent with our model within $\sim
1$-$2\sigma$. Although our error bars are similar to those of the 2014
measurement, our method is significantly more robust to CIB contamination. Our
moment-deprojection approach lays the foundation for future measurements of
this signal with higher signal-to-noise maps from ground-based telescopes,
which will precisely probe the astrophysics of the intracluster medium of
galaxy groups and clusters in the intermediate-mass ($M\sim 10^{13} -10^{14}
h^{-1} M_\odot$), high-$z$ ($z<\sim1.5$, c.f. $z<\sim0.8$ for the tSZ
auto-power signal) regime, as well as CIB-decontaminated measurements of tSZ
cross-correlations with other large-scale structure probes.
