Originally Posted by
dabac
Me, I'd remove the axle adjustment screw, grind a V-shaped groove along the crack on both sides. Stick an axle sized bolt at the open end of the dropout, use some washers and a nut to maintain alignment. Then use whatever arc welder that was handy to run a bead through the ground out groove to fill it up. A quick swipe with an angle grinder to get the dropout face flat and call it done. Maybe a dab of primer.
With luck, the adjustment screw would still fit. If it didn't, I'd try to retap it. If that didn't work I'd just consider it a "war wound" and do without it.
This is a plausible repair option. Either welding as you suggest, or brazing ought to work. I'd get rid of the adjuster bolt completely, though, and fill the hole with brass so it won't act as a stress raiser again. You don't really need those adjusters, after all; they're just a convenience in getting the wheel centered and the holes they require in the dropouts are a common failure point as the OP has discovered.