Can this crack in the steel chain-side dropout be welded?

Yes I think it can be repaired as the failure is of the dropout itself which is quite thick steel (about 4 or 5mm).

It needs the paint removing an inch or so out, to be cleaned up, beveled and carefully TIG welded. I have repaired a very similar failure like this before although that was a little bit easier as the break was a bit further away from the end of the chainstay.

The chainstay is brazed into the dropout so we don't know for sure if it is a weldable alloy. But it probably is (it is most likely 4130 cromoly). If it's welded at the BB end that proves it is. There's therefore no harm in the repair tying in to the end of the chainstay a bit.

As Carel said there is a risk of the existing braze melting out and ruining your day as it's so close. But still worth a try I'd say.

Replacing the whole dropout is quite tricky because you'd need to find one the same size and then try and unbraze the existing one without damaging the ends of the stays. Or you could make a slightly bigger one so you can cut the stays back a bit. Either way you also need a jig or fixture to hold the dropout in the right place. All this would be quite a bit more work so probably would not be economical.

Having said all this I can't guarantee that whoever you take it to will want to have anything to do with it or how much he would charge. It wouldn't take more than about 20 minutes to do.

You need to consult in person with someone local to you who can make a qualified recommendation.

"It is Steel so can be welded" is mostly true, but there are always exceptions. A good welder will give you their professional advice. Don't bother asking someone with a stick welder in their garage.

I've personally just got a quote on welding a crack in a seat tube, and that came in at $100 NZ. The bike cost me half that to buy, but I can't replace the frame for anything like that.

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When you do find someone local, prepare the frame by stripping everything off the bike. Everything that is not frame needs to be removed, and you should cover and protect all the threads/bearing surfaces with rags and tape.

Yes, I would remove the fork and headset, and the bottom bracket, leaving just Frame and Paint.

The only non-frame thing I might leave in there would be an old scrap axle bolted into the dropouts, to help keep everything in line and not getting extra-stressed while manhandled.

Also, clean the bike. Noone likes getting dirty, and your welder is more likely to agree if its not filthy.

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If the welder suggests unbrazing and replacing the dropout instead of welding, then you should take that advice. They may suggest a sleeve or fishplate for strength — take that advice. Figure out if it will interfere with your chain/cassette or fitting of the rear wheel.

And lastly, if its not safe to fix, or if its super expensive, consider retiring the frame and put those costs toward another bike.

A lifetime frame warranty may not require that you have the receipt.
Take it to a Trek dealer and see what they say.

Several of the shops I worked at did frame warranty work with no receipt.

I'm a TIG welder.

The break is thru the drop out not the brazed joint The dropout could welded and re-brazed into the frame, but better to make or get a new dropout and braze that in. Best option is see a framebuilder in your area or cheaper option would be a engineering shop that does brazing and welding..

Yes, I would TIG weld it but insert a small piece of flat steel that would go into the tube and lay flat on the rear wheel yoke by about 25mm then weld all round.

As a welder who has extensive experience with thin wall tubing. This could easily be welded with TIG. Depending on the metals just a little Beveling and alignment of the stay in a jig and you’ll be all set.

I'm also a combination welder, MIG or TIG would be a good choice as long as the parent metals are throughly cleaned and free of paint and dirt. As for the inspection of all other welds the use of die penatrant and developer is the easiest and fastest way to check for cracks. I use this method or race car chassis's after drivers have had a hard impact.