The 12-team tournament, which had originally been scheduled to take place from June 12 to July 12 in Colombia and Argentina, follows the Euro 2020 championship in being pushed back by a year with global sport being brought to a virtual standstill by the coronavirus outbreak.
"This is an extraordinary measure for an unexpected situation and responds to the fundamental need to avoid the exponential evolution of the virus," said CONMEBOL president Alejandro Dominguez.
Dominguez said the rescheduled tournament would be played between June 11 and July 11, 2021.
"It wasn't easy to take this decision but we must safeguard at all times the health of our players and all those who form part of the big South American football family. Have no doubts that the oldest international tournament in the world will be back stronger than ever in 2021."
The Copa America is one of the biggest draws on the football calendar, with players such as Lionel Messi, Neymar, Sergio Aguero, Luis Suarez, and James Rodriguez all expected to take part.
The Euro 2020 finals, which was due to take place over the same dates across 12 countries in Europe, was also postponed for a year.