The former University of Otago vice-chancellor, who continues to undertake research at the Dunedin campus, will assume his new role on July 1.
Prof Skegg will succeed Dr Garth Carnaby in the role, which has a term of up to three years.
Prof Skegg said the invitation to become society president was a surprise and an honour.
"The Royal Society is an excellent organisation. This will be a fascinating role and I am pleased to be able to contribute from my base in Dunedin," he said.
Prof Skegg, a distinguished cancer researcher who led the university from 2004-11, chairs the Government's Science Board as well as a World Health Organisation committee on reproductive health and research.
Auckland-raised Prof Skegg studied medicine at Otago, then travelled on exchange to Harvard University before being awarded a Rhodes Scholarship to Oxford.
He returned to Otago in 1980 when he was appointed to the departmental chair in Preventive and Social Medicine. He was awarded an OBE in 1990 and knighted in 2009.
Society academy chairman Dr Stephen Goldson said it was appropriate Prof Skegg was a Fellow of the society, based on his distinguished career.
"Coupled with this, he has had extensive experience in leadership and governance that will hold him in good stead," Dr Goldson said.
University of Otago vice-chancellor Prof Harlene Hayne said Prof Skegg was an excellent choice, who "brings a wealth of talent and experience to the role.
Our university flourished under his leadership style and I fully expect his outstanding scholarship and collegial style will have a similar influence on the society's future growth and direction."
The Royal Society is an independent national academy of science, technology and the humanities. It is a federation of scientific and technological societies and individual members forming an educational and research charity which operates under a parliamentary Act for the advancement and promotion of science, technology and the humanities.
The society represents nearly 20,000 scientists, scholars, technologists and technicians.
