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Consider a foliation in the projective plane admitting a projective line asthe unique invariant algebraic curve. Assume that the foliation is generic inthe sense that its singular points are hyperbolic. We show that there is aunique positive ddc-closed (1,1)-current of mass 1 which is directed by thefoliation and this is the current of integration on the invariant line.  A unique ergodicity theorem for the distribution of leaves follows: for anyleaf L, appropriate averages of L converge to the current of integration on theinvariant line. This property is surprising because for most of such foliationsthe leaves (except the invariant line) are dense in the projective plane. Soone could expect that they spend a significant amount of hyperbolic time inevery open set and that there should be a fat ddc-closed non-closed currentwith support equal to the projective plane.  The proof uses an extension of our theory of densities for currents.Foliations on compact Kaehler surfaces with one or several invariant curves arealso considered.

Unique ergodicity for foliations in P^2 with an invariant curve