Atlanta signed Weems to a minor-league contract Tuesday, Spencer Nusbaum of The Washington Post reports.
Weems cleared waivers Friday and was assigned to Triple-A Rochester.
The Nationals designated Weems for assignment Tuesday, Jessica Camerato of MLB.com reports.
The Nationals reinstated Weems (lower leg) from the 15-day injured list Saturday, Spencer Nusbaum of The Washington Post reports.
The Nationals placed Weems on the 15-day injured list Friday with right shin splints, Mark Zuckerman of MASNSports.com reports.
Weems has posted a 5.68 ERA, 1.55 WHIP and 32:19 K:BB through 38 innings with one win and four holds in 37 appearances.
Weems (1-0) recorded a blown save and a win Tuesday against the Giants, giving up two runs (one earned) on three hits in an inning of relief. He struck out one.
Weems (illness) struck out a batter and worked around a walk while allowing no runs over two-thirds of an inning of relief en route to notching his first hold of the season in Wednesday's 5-3 win over the Pirates.
Weems wasn't available to pitch Monday due to illness, Mark Zuckerman of MASNSports.com reports.
Weems has posted a 10:3 K:BB without allowing a home run over 9.2 Grapefruit League innings this spring.