  We extend Struwe's result (Acta Math., 1988) on the existence of free
boundary constant mean curvature disks to almost every prescribed boundary
contact angle in $(0, \pi)$. Specifically, let $\Sigma$ be a surface in
$\mathbb{R}^3$ diffeomorphic to the sphere, and let $\Sigma'$ be a convex
surface enclosing $\Sigma$. Given $\tau \in (-1, 1)$ and a constant $H \geq 0$
below the infimum of the mean curvature of $\Sigma'$, we show that for almost
every $r \in (0, 1)$, in the region enclosed by $\Sigma'$ there exists a
branched immersed disk with constant mean curvature $rH$ whose boundary meets
$\Sigma$ at an angle with cosine value $r\tau$. Moreover, the constant mean
curvature disks we construct have index at most $1$.
