  In this article, we solve the constant mean curvature dirichlet problem on
catenoidal necks with small scale in $\mb{R}^3$. The solutions are found in
exponentially weighted H\"older spaces with non-integer weight and are a-priori
bounded by a uniform constant times $r^{1 + \gamma}$, where $r$ denotes the
distance to the axis of the neck and where $\gamma$ belongs to the interval
$(0, 1)$. By comparing the solutions with their limits on the disk, we improve
the estimate to $\gamma =1$. As a corollary, we prove differentiability of
solutions in $\tau$ down to $\tau = 0$. The surfaces we construct have
applications to gluing constructions.
