  Given a two-dimensional Haag-Kastler net which is Poincar\'e-dilation
covariant with additional properties, we prove that it can be extended to a
M\"obius covariant net. Additional properties are either a certain condition on
modular covariance, or a variant of strong additivity. The proof relies neither
on the existence of stress-energy tensor nor any assumption on scaling
dimensions. We exhibit some examples of Poincar\'e-dilation covariant net which
cannot be extended to a M\"obius covariant net, and discuss the obstructions.
