  In this paper we consider the one-phase Stefan problem with surface tension,
set in a two-dimensional strip-like geometry, with periodic boundary conditions
respect to the horizontal direction $x_1\in\mathbb{T}$. We prove that the
system is locally null-controllable in any positive time, by means of a control
supported within an arbitrary open and non-empty subset. We proceed by a linear
test and duality, but quickly find that the linearized system is not symmetric
and the adjoint has a dynamic coupling between the two states through the
(fixed) boundary. Hence, motivated by a Fourier decomposition with respect to
$x_1$, we consider a family of one-dimensional systems and prove observability
results which are uniform with respect to the Fourier frequency parameter. The
latter results are also novel, as we compute the full spectrum of the
underlying operator for the non-zero Fourier modes. The zeroth mode system, on
the other hand, is seen as a controllability problem for the linear heat
equation with a finite-dimensional constraint. The complete observability of
the adjoint is derived by using a Lebeau-Robbiano strategy, and the local
controllability of the nonlinear system is then shown by combining an
adaptation of the source term method introduced in \cite{tucsnak_burgers} and a
Banach fixed point argument. Numerical experiments motivate several challenging
open problems, foraying even beyond the specific setting we deal with herein.
