I've had very good results with the frames I've dipped in oa solution. But it's a big pain in the butt, especially here in Vermont in the winter, when I can't do it outside and my marital situation precludes my doing it inside. How can get good results without dipping? I know that some people use super-fine steel wool (not sure how many 0s) and WD40 or something similar. But this is an old Peugeot (it's the same bike in the "geezering" thread of a while back) and given the tenderness of Peugeot paint, I hesitate to use anything abrasive on it at all.
I should say that the frame isn't REAL rusty--some surface rust on the top tube, around the bottom bracket shell, probably inside the shell, etc.
What suggestions to people have?

try naval jelly or rust converter. both work great for outside surface rust. naval jelly will eat away the rust, converter turns it into a primer-like black coat. whichever works best for you

or as a super cheap method, aluminum foil, vinegar, old rags.

It seems to me that all of your suggestions will beat the daylights out of the existing paint. Or is that not correct?

I don't think so. I've used rust converter and it doesn't affect paint at all. Naval jelly is strong, so it might. I'd search for phosphoric acid on BF to make sure.

Vinegar and old rags shouldn't hurt the paint. Dilute a bit if necessary, but I find most vinegar is dilute enough. Aluminum foil I just tend to concentrate around rusted bare steel - twist it into a point and rub. It will take off the paint if you rub it at a painted spot.

How about wrapping the affected areas with rags and saturating with an OA solution? Just go back and saturate every once in a while. Just be sure to have something underneath to catch drips.

+1 Rags soaked in OA solution.

That sounds like a good idea too. I've never tried that myself, but it makes sense.

Plus, Thrifty Bill says it works. I respect him on OA as much as I respect Khatful on polishing.

EDIT::to make this clear, this means I respect him on OA a lot

Okay, the wrap-the-frame-in-OA-soaked-rags idea suggests this variant: how about wrapping whole frame in strips of t-shirt fabric, or something like that--kind of a time-consuming process, obviously--so it looks like an Egyptian mummy? Stuff some fabric in the bottom bracket shell, too. Then wet all the cloth down with white vinegar and seal the whole thing in a plastic garbage bag for a couple of days so it doesn't dry out. Then unwrap everything (another PITA) and see how it looks. Neutralize with baking soda. Dry off.
OA would obviously be another option, but I imagine that un-mummifying the frame would get OA all over one's hands and clothes. I'd worry less about getting spattered with vinegar.
I think I may try that and report back, assuming I can find enough old T-shirts.

It would probably work. Pick up a few rolls of non-sterile medical gauze and use that for the wrap.

Woo Hooo! This is some of the best thinking on rust removing that I’ve seen. Next time I have a somewhat rusty lugged frame, I’m gonna drop some poly on the garage floor and wrap the rusted areas in rags or gauze then drizzle some oxy water on the wounded areas. Beats the heck out of the tub method I’ve been using.

I'd thought about gauze, but it seems pretty insubstantial. It doesn't seem like it would absorb much solution. What's needed is something like absorbent cotton in a ropy kind of format. Is there any such thing? I don't know. Tree nurseries use burlap that comes in strip form to wrap the trunks of young trees. It's probably pretty cheap and available in long rolls. I wonder how that would work?

EDIT: Just checked a nursery supplier--they sell 100-yard rolls of 4-inch burlap for about $9. That sounds pretty promising.

Make a paste out of Barkeeper's Friend, and apply for 1-2 minutes. Wipe off. If needed, apply again. Very easy. It's a low strength solution, and good for applied use.