Alas, it had not.
But Welsh rarebit can also be made with milk or so said the old copy of Mrs. Beeton's Cookery Book that I had picked up years ago in the UK. (How old? I'm not sure as there's no copyright date. A little quick Internet research leads me to believe it might have been published as long ago as 1911.) I've enjoyed looking through this book from time to time, but I had never made a recipe from it. Since one of the secondary goals of this year's blogging project is to weed through all the cookbooks I've accumulated, this was a good opportunity. Following is the recipe as written.
Welsh Rabbit or Rarebit
Ingredients - 4 oz. of Cheshire or Cheddar cheese, 1/2 oz of butter, 2 or 3 tablespoonfuls of milk or ale, mustard, pepper, buttered toast.
Method - Cut the cheese into small pieces, place these in a saucepan with the butter, milk, or ale, 1/2 a mustardspoonful of mustard, and pepper to taste, and stir the mixture by the side of the fire until it resembles a thick cream Have ready some squares of hot, well-buttered toast, pour on the cheese preparation, and serve as quickly as possible.
NB - I halved the recipe, used 2 T milk, and 1/2 t of dried mustard. The result was very thick.
Would I make this recipe again? Yes, but I'd be more careful about heating up, because the sauce broke a bit at the end. Upon consideration, I don't much care for the cracked pepper bread.