0 MT

U J,
I'm not clear what stage you're at. Does your little lathe let the small end of an M0 into it's sockets or are you, like me, unable to even start the standard M0 into the hole. Neil called the Adept a M00 and thats not a bad definition.
If Adept is what you've got then taper turning is probably involved but if you can get an M0 stubby into the headstock spindle taper you can proceed from there. cut off the excess leaving enough to turn a screwed spigot, say M4 or M5 on the end.
Now make a screwed bush out of say 15mm dia bar, matching your M4/M5 and rough turn to size for your drill chuck. To get best concentricity mount the part finished bush on the stubby arbor, put it back in your headstock spindle and turn the remaining dias true. Die the thread for your chuck, assemble it and go drill some holes.
I would not do any welding, too many stresses and distorsion. All normal cutting tends to tighten the threads up so there is little chance of things coming loose unexpectedly.
If you are stuck wiith lathe that has no facility for offsetting the tailstock, like the Adept, and want to do a slow taper there are simple attachments to put an adjustable centre on the tailstock barrel. these often use the MT hole but can also mount straight onto the outside diameter of the barrel. Making from scratch can be a bit involved but simpler designs should be possible with little more than a lathe with chuck and a pillar drill.
The frustrating part is setting it up to accurately match an existing, M00 say, taper

Bob D