While the draw took place on Tuesday, the nations already knew nearly all of their opponents with teams split into zones for next year's tournament in Argentina and Colombia.
Eight teams will qualify for the quarter-finals with the final to place on 12 July in Barranquilla, Colombia.
In Group A, co-hosts Argentina are joined by several other South American powerhouses.
Messi will be a star attraction for 14-time winners Argentina, who finished third at this year's Copa America, and he will enter the tournament as the reigning men's Ballon d'Or victor, an award he has claimed on a record six occasions.
Qatar played in the Copa America 2019 and despite losing two of their three matches, proved a tough team, ensuring that they won't be a surprise package in their second appearance in the upcoming matches in 2020.
Notably, Argentina last won the Copa America title in 1993 and since then they have finished as runners up four times in 2004, 2007, 2015 and 2016.
The official 2020 Copa America ball, courtesy of Nike.