Will a 27-inch x 1-1/4-inch tire fit on a 27x1 1/8 rim

If you mean your rim has a 27-1-1/8" tire then yes.
As long as it clears the frame/stays etc., else it may not fit the bike.

Thank you very much, Bill

Just for curiosity, you might state the type of bike.
If it were some "exotic" race bike of yore, there could be possible exceptions.
If it's a fairly plain vanilla 2X5 or 2X6 speed from the 70-80's, it might have actually came with 1-1/4's.
And if it's TUBULAR, I'm ignorant of that category to comment.

Bill, I have my bikes in the gallery. My venture is done, I love this bike, working on the 15-speed

If it's tubular, it's likely 700C, not 27" anyway (unless more exotic 650C or 650B). Look for ETRTO size markings on the tire and/or rim. nn-630 indicates 27"; nn-622 is 700C, nn-671 is 650C, nn-584 is 650B.

But yes, a 27" x 1-1/4" (32-630) tire will fit on a 27" (630mm bead seat) rim. Frame clearance may be an issue, but that's another matter.