The tie will be played on Saturday 2 July at 15:00 BST in Portlaoise.
Laois brought on seven replacements at O'Moore Park, one more than permitted.
The Central Competitions Control Committee released a statement on Tuesday night, adding that Laois had accepted the proposal.
Shane Murphy came on as the seventh substitute in added time as Laois took a 1-10 to 0-10 win in Portlaoise.
Conor Meredith scored the decisive goal and Laois were subsequently drawn to face Clare in the second round of the qualifiers.
However, Murphy's introduction as a black card sub for Brendan Quigley could prove costly for the Leinster side if Armagh were to triumph in the re-playing of the game.
Saturday 2 July
Round 1A All-Ireland SFC qualifier refixture
Laois v Armagh, 15:00 BST, O'Moore Park, Portlaoise
Round 2A
Sligo v Leitrim, 18:00 BST, Markievicz Park, Sligo
Saturday 9 and Sunday 10 July
Round 2A
Clare v Laois/Armagh, Cusack Park, Ennis, Time tbc.
Derry v Dublin/Meath, Details tbc.
Cavan or Tyrone v Carlow, Details tbc.
Armagh are set to replay their All-Ireland qualifier Round 1A game after Laois were found to have used too many substitutes in their victory over the Orchard County on Saturday.