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.

Scotland goes to the polls to elect its local councillors on 4 May.
The Lib Dems are putting forward candidates in a record number of wards, aiming to build on recent by-election wins.
Party leader Willie Rennie said Lib Dem candidates would "put people first", accusing the SNP and Greens of "putting independence first".
The Lib Dems are looking to bounce back from the 2012 elections, where they lost more than half their council seats in the aftermath of a coalition deal with the Tories at Westminster.
The manifesto describes the party as "intent on winning more seats and playing a major part in Scottish councils".
While campaigns in different council areas will highlight local issues, the manifesto sets out the party's national priorities, including mental health, education and scrapping the council tax.
It also underlines the party's backing for a reformed federal UK and "continued membership of the European Union", proposing a referendum on the terms of the Brexit deal.
At the launch event in Inverkeithing, Mr Rennie said he was confident the party could build on by-election wins and gain seats across Scotland.
He said: "Local Liberal Democrat councillors have an impressive record of action and service for their communities. They work all year round for local people, not just at election time.
"This year's elections present an excellent opportunity to build on our by-election successes across the country with gains from all parties. We are contesting more wards and looking to win more seats on 4 May.
"As the SNP and Greens put independence first, Liberal Democrats will put local people first. Scotland does not need another divisive independence referendum."

A small group of people ran on to the M4 spur road and laid down in front of oncoming traffic on Saturday, causing temporary disruption.
People aged between 21 and 67 have been charged with wilful obstruction of the highway, according to the Met Police.
They have all been bailed to appear at Ealing Magistrates' Court on 22 December.
Among the accused are seven Londoners. They are: Isabelle Anderson, 30, of Stratford; Madeleine Allis-Petersen, 24, of Ealing; Joanne Louise Bodimeade, 28, of Lambeth; Alexis Delage, 25, of Lewisham; Sophia Lysaczanko, 28, of Haringey; Tom Venner-Woodcock, 29, of Southwark; and Tess Lotter, 30, of Camden.
The others charged are: Antoine Thalmann, 25, and Henry Owen, 23, both of Oxford; Simon Bramwell, 44, of Stroud, Gloucestershire; Ian Bray, 49, of Kirklees, West Yorkshire; Graham Lewis, 53, of Wells, Somerset; Thomas Harford, 26, and Margaret Charnley, 67, both of Bristol; and Sibi Moore, 21, of Sidmouth, Devon.
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.
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The Scottish Liberal Democrats have launched their national manifesto for May's local authority elections.
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Fifteen people have been charged after campaigners against airport expansion staged a protest near Heathrow Airport.