Wexford's consolation prize for a heavy defeat to Kilkenny in their Leinster semi-final is a round One qualifier meeting with Jimmy Barry-Murphy's Cork.
Liam Dunne's men will entertain the Rebels at Wexford Park and will need to replicate the form which saw them defeat Clare in last year's qualifiers.
The two sides who suffered heavy defeats to Galway in the Leinster championship, Dublin and Laois, will meet in Portlaoise.
Davy Fitzgerald's Clare will have home advantage when they take on Brian Whelehan's Offaly who are still reeling from their defeat to Laois.
Yesterday's Munster semi-final losers Limerick will have to overcome Westmeath in Mullingar if they are to keep their All-Ireland hopes alive.
All-Ireland SHC Hurling Round One qualifier draw:
Laois v Dublin
Westmeath v Limerick
Clare v Offaly
Wexford v Cork
Confirmation of home venues, times and all dates for the fixtures will be confirmed following a meeting of the CCCC on Monday afternoon.