  We consider simply connected log-Riemann surfaces with a finite number of
ramification points. We prove that these surfaces are biholomorphic to C with
uniformizations given by entire functions of the form F (z) = \int Q(z)
e^{P(z)} dz where P, Q are polynomials of degrees equal to the number of
infinite and finite order ramification points respectively. Conversely any such
entire function defines a simply connected log-Riemann surface with finitely
many ramification points.
