If you want to do the correction correctly, rotten, you hit the nail on the head with your concrete suggestion. Tell Unc his foam idea won't fly, as most foams are not meant to be watertight. But you need to do the prep and formwork for your concrete very carefully, as mistakes will only make the repair a source of future problems.
Use a small chipping hammer with chisel bit, and chip out every last bit of unsound material (rotten mortar, concrete, stone, etc.) that the new concrete will be bonding to. Leaving unsound material in place will only ensure that your patch will fail, and eventually have to be completely chipped out and redone. Build the solid-side form (basement side) of plywood on a wooden frame of 2 x 4s, screwed into drilled anchors against the adjacent wall faces and braced with a kicker or two to keep it from bowing.
Then build your outside form with a hopper configuration at the top 6" or so to enable you to get concrete into it. Build it with the top a few inches higher than the top of the hole you're filling, to enable you to build a head of wet concrete inside the form, up to the top of the opening. Some similar screwed anchors on the sides and bottom will hold it in place, along with a few kickers as well. Do not install it yet, but do a dry run to make sure it can be quickly slapped into place.
I'd suggest using pre-bagged concrete, but something with more strength than the stuff used to set fence posts. I've had good results with Quikrete 5000, which is rich enough in cement to do what's needed for your application. You could hand-batch it in a wheel barrow, just outside where you will be dumping it into your form hopper. Use a "loose" mix, so it will flow. But here is the important part--mix up a Portland cement slurry (consistency of heavy cream) to use as a bonding agent, and brush it onto all of the prepared concrete surfaces that you've previously made damp by baptizing with water, using a mason's brush. You don't want the bonding agent to dry before placing concrete against it, so time is of the essence. Slap on the outside form after wetting both forms, and dump the mixed concrete into the hopper as quickly as you can while thoroughly consolidating the concrete (I'd suggest borrowing or renting a spud vibrator if you don't own one, or at least vigorously beating on both sides of the formwork with hammers to eliminate all of the entrapped air). You want to keep the hopper filled to the top while vibrating the fluid mix, making every effort possible to get the concrete to flow to the far top corners and back edge of your opening.
Strip the forms the next day, such that the concrete will be green enough to easily chip the excess off at the widened form area. Be gentle with your chipping hammer, always breaking away from the center, as you don't want to crack and ruin the "keeper" concrete. Mix up some fluid mortar to trowel over the exposed roughened concrete face (again using some bonding agent), and use it for filling any voids present around the edges of the patch (particularly at the top).
As you can probably tell, this operation is best done with a helper or two.