It's very likely you're thinking of the 1956 film "The Swan", starring:
* Grace Kelly as Princess Alexandra
* Alec Guinness as Dr. Nicholas A. Tarlen
* Louis Jourdan as Prince Albert
* Agnes Moorehead as The Grand Duchess
* Leo G. Carroll as The Grand Duke
This film was a big hit, showcasing Grace Kelly's elegance and acting talent. It's a romantic drama that follows a princess's love story with a prince and a doctor who is entangled in the royal family.