An anchor assembly for use in joining together a masonry wall structure and a supporting back wall structure wherein the back wall structure is preferably formed using insulated concrete form (ICF) blocks each having a pair of opposed ICF panels associated therewith, the anchor assembly including an anchor member and a tie member. The anchor member includes an anchor portion adapted to penetrate one of the ICF panels and having at least one opening associated therewith so as to lie between the opposed ICF panels when attached thereto for allowing poured concrete to flow therethrough, and an attachment portion having a tab member positioned and located to overlay an ICF panel flange member when the anchor member is inserted within the ICF panel, the tab member being attachable to the ICF panel flange member. The attachment portion further includes an elongated slot for receiving the tie member, the tie member being adjustably movable along the length of the slot so as to be positionable between the courses of brick or other masonry material associated with the masonry wall structure as the masonry wall structure is being erected.