I have a 1903 rifle made by Remington in 1942 with the older leaf rear sight. The sight adjusts left and right all right, but is loose in its mount. I guess that this is due to fitting on the collar (on the stud that sits on top). Is there a way to tighten this up?

Not quite sure just where you are describing the looseness, but the answer is yes, they can be serviced or even replaced. Are you speaking of the pin on which the ladder pivots or elsewhere? You can do a search(Google, etc.) and find replacement parts/sights for the '03 rear sight. Sure someone else can give you clearer answer than I but the problem can be corrected. Might mention that the groove in which the sight travels for windage can be tightened up by careful peening of the upper surface, just a thought.

I believe he's speaking of the stud that is on the entire sight base. By removing the windage screw adjustment, you can remove the entire sight leaving the base with that stud I'm
talking about. Am I correct original poster?

That windage screw adjustment have a spring on it that keeps the sight tensioned.

SmokinJ, you are correct, when I remove the whole rear sight from the collar, I can see the cylindrical stud that the sight pivots on. The problem is that the rear sight assy. fits loosely on the stud and wobbles left and right, which isn't good for a rifle sight. I did get another rear sight from Numrich, which fit way too tightly on the stud. A little help from the Dremel on the inside of the "new" rear sight and it is again loose (drat), just not like the first one.

Well that sure sucks. Little too much with the Dremel. You know what they say measure twice cut once. I've had quite a few 03's but never had that problem. In fact just got one in which pieces of the rear sight were missing and I too got parts from Numrich. They fit just fine. I wouldn't mess with the stud because if you mess that up the rear sight base has to be replaced and it's not "real" easy to get off. If you get another one and it's tight (which means to me it goes on the stud but is very hard to move) maybe a little oil or grease on it and work it back and forth some to wear it in. If its more then that just polish the hole don't use a Dremel grinding tool, use very fine sand paper or Crocus cloth.

"Out of the box..." thought, but could you utilize MarineTex to obtain the correct fit of the female/male system?? Once cured that stuff can be machined/threaded, etc. Apply fair amount of floor wax for release agent, use C clamp to hold together for cureing time and see what happens.

After shooting this rifle again last week, I went back to the original rear sight assembly and found my solution. I cut a piece of thin plastic to act as a shim on top of the stud and under the rear sight. Works perfectly; sight does not wiggle and can still be adjusted.