Connie was born in Copenhagen, Denmark but she was raised in a small seaside village named Elling near Frederikshavn in Denmark. She began her acting career working alongside her mother on the local revue and variety scene. When she was 18, she travelled to Paris, France where she worked as an actress and model. Her work in Paris led to further work and study in Italy — at drama school in Rome and il Piccolo Teatro di Milano in Milan. She lived in Italy for many years, before she moved to the United States, where she still resides. Her son Sebastian was born in 1990 and she lives with him in Tiburon, California.