Hello there!
Seems you and I have the same thoughts - building a double rifle.
It can be done, however I would not recommend you doing it unless you got someone close by thta can help you out. Now, don't think I am bashing your idea, nor telling you that you can't do it - it's just that what you need to get it done will probably run you as much as buying a good quality used one (as long as you steer clear of Purdeys which tend to be a bit more costly...) Because you need a lathe, mill, good workingplace, a truck load of good quality files and a LOT of time!

I never built one myself as of yet, but have seen several being built on Merkel model 8's, 20 and 16ga, converting them to 8x57JRS. Process is quite forward:
1) Chop off the barrels at the front end of the chamber.
2) Ream the old chambers mirror smooth.
3) Turn down 2 barrels (better get barrels ment for double rifles from the beginning or yoy will end up with a very front heavy gun) so that their aft ends fits Exactly inside the reamed out chambers and protrude about 1/16" to the rear.
4) Soft solder the barrels in place (Yes, soft solder will do just fine - Do Not Weld Barrels!)
5) Solder the newly made ribs in place, including front sight, fore end lock notch and anything else you migt wanna have, like the front wedge and a quarter rib and so on. Ooops, forgot to tell that all soldering done must be nothing short of Perfect.
6) File the aft en absolutely square to the action
7) Make new extractor cos the old one won't work
8) Fit new extractor
9) AFTER fitting new extractor - cut chambers with extractor in place (it's hard to ream them chambers b4 you solder the barrels in place. So to do this you need a good quality drill press and the knowledge on "how to")
10) Test fire (if you dare)
11) Regulate rifle cos it surely won't shoot to the same POI as it is your first gun.

There are a few more minor details to take into account, like that you can't ream the old chambers parallel but they must actually be reamed at an angle so that the axis of them 2 bores wil meet at some distance. You also will want to file the outside contour of the new barrels and a few more things. Oh, almost forgot to say that you need to get new firing pins in place (smaller diameter) than the old ones cos of the increased pressure. It does take some time and some skill not to FUBAR.

There is another timeconsuming way to go too: Make a completely new set of rifled barrels for your shotgun-of-choice. Then you just pick the right set of barrels regarding what you want to hunt. For pigeons go with them shotgun barrels, for large game you just place your double rifle barrels in place. Nice!

(Heck, the more I think about it the more I realize that I should not spend time in my shop building my own double rifle but put in extra hours at work instead so that I can just buy myself one!

Nah... I kinda look forward to it!
Friends that has done all this spent bout 5 months - 8 hours a day 5 days a week b4 they were finished. Then it was first time projects. On the other hand they got themselves some nice double rifles!)

So there you go; A short guide on how to build a double.

Be safe

Per