The extraordinary meeting was convened for Council members to discuss the presidential succession, following the withdrawal of Professor David Oliver as president elect last week.
Dr Clarke - who will become the sixth Royal Papworth consultant to become a president of a professional membership body - should assume the role in September for a four-year term.
Current RCP president Sir Andrew Goddard said: "I was very sad when David withdrew from the role of president but am grateful that the election process allowed us to elect Sarah with a clear mandate from the fellowship. The RCP Council strongly supports her election and we are delighted to have her at the helm of the college over what are likely to be very challenging times."
Dr Sarah Clarke, Consultant Cardiologist at Royal Papworth Hospital NHS Foundation Trust, RCP Clinical Vice President and past president of the British Cardiovascular Society, said: "I was delighted to receive the call to serve as president. It is a huge privilege to be the president of the RCP and I want to thank the fellowship and Council who have put their faith in me and will be by my side as we navigate the challenging times ahead for our profession.
"I also want to thank my colleagues at Royal Papworth Hospital, the wider NHS, and my friends and family for their never-ending support over many years which has enabled me to get to this position.
"The RCP has found itself in a unique situation with this election. I have the utmost respect for my colleague and friend David Oliver, who has put his health and wellbeing first in making a difficult decision to step down from the presidency. I will be doing this for both of us."
Dr Sarah Clarke announced as new RCP president - Full RCP release