  Let C and C' be two smooth self transverse immersions of S^1 into R^2. Both C
and C' subdivide the plane into a number of disks and one unbounded component.
An isotopy of the plane which takes C to C' induces a 1-1 correspondence
between the disks of C and C'. An obvious necessary condition for there to
exist an area-preserving isotopy of the plane taking C to C' is that there
exists an isotopy for which the area of every disk of C equals that of the
corresponding disk of C'. In this paper we show that this is also a sufficient
condition.
