No, not uniquely. Conservation of angular momentum is a necessary condition, but it is not sufficient.
Kepler's First Law says that the planets orbit in elliptical paths with the Sun at a focus of the ellipse. This specifically depends on
- an inverse-square law force and
- a negative total mechanical energy, with the reference zero for the potential energy is infinite separation distance.
This force and this energy are not dictated by conservation of angular momentum. Conservation of angular momentum results for any form of a central (aka, radial) force.
If the force is repulsive or the energy is too large, the orbit will not be elliptical. The first case would happen for like-signed charges orbiting each other (not planetary motion), and the second, a comet which executes a parabolic or hyperbolic orbit. These systems conserve angular momentum, but definitely don't follow Kepler's elliptical law., not