  In this paper we give a complete characterization of the scaling limit of the
critical Interacting Partially Directed Self-Avoiding Walk (IPDSAW) introduced
in Zwanzig and Lauritzen (1968). As the system size $L$ diverges, we prove that
the set of occupied sites, rescaled horizontally by $L^{2/3}$ and vertically by
$L^{1/3}$ converges in law for the Hausdorff distance towards a non trivial
random set. This limiting set is built with a Brownian motion $B$ conditioned
to come back at the origin at $a_1$ the time at which its geometric area
reaches $1$. The modulus of $B$ up to $a_1$ gives the height of the limiting
set, while its center of mass process is an independent Brownian motion.
  Obtaining the shape theorem requires to derive a functional central limit
theorem for the excursion of a random walk with Laplace symmetric increments
conditioned on sweeping a prescribed geometric area. This result is proven in a
companion paper arXiv:1709.06448.
