  introduce {\sc Planar Disjoint Paths Completion}, a completion counterpart of
the Disjoint Paths problem, and study its parameterized complexity. The problem
can be stated as follows: given a, not necessarily connected, plane graph $G,$
$k$ pairs of terminals, and a face $F$ of $G,$ find a minimum-size set of
edges, if one exists, to be added inside $F$ so that the embedding remains
planar and the pairs become connected by $k$ disjoint paths in the augmented
network. Our results are twofold: first, we give an upper bound on the number
of necessary additional edges when a solution exists. This bound is a function
of $k$, independent of the size of $G.$ Second, we show that the problem is
fixed-parameter tractable, in particular, it can be solved in time $f(k)\cdot
n^{2}.$
