  Two-pointed quantum disks with a weight parameter $W > 0$ are a family of
finite-area random surfaces that arise naturally in Liouville quantum gravity.
In this paper we show that conformally welding two quantum disks according to
their boundary lengths gives another quantum disk decorated with an independent
chordal $\mathrm{SLE}_\kappa(\rho_-;\rho_+)$ curve. This is the finite-volume
counterpart of the classical result of Sheffield (2010) and
Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed
quantum surfaces called quantum wedges, which is fundamental to the
mating-of-trees theory. Our results can be used to give unified proofs of the
mating-of-trees theorems for the quantum disk and the quantum sphere, in
addition to a mating-of-trees description of the weight $W =
\frac{\gamma^2}{2}$ quantum disk. Moreover, it serves as a key ingredient in
our companion work [AHS21], which proves an exact formula for
$\mathrm{SLE}_\kappa(\rho_-;\rho_+)$ using conformal welding of random surfaces
and a conformal welding result giving the so-called SLE loop.
