Yes you could, it would be called an "ANY OTHER WEAPON" (AOW).
You would need to add a forward pistol grip, and it would be exempt from a minimum length barrel requirement AND you could have all of the evil features that the 94 crime bill prohibits.
You will need to pay the BATF a $200 manufacturing tax and then subject yourself to an FBI background check, plus you would need your local LEO to sign off on it.
You could never put any type of a stock on it again though.
You could build it as a Short Barreled Rifle, but you can not remove the stock.
Same fees as above apply.
27CFR, VOL I, Parts 1 - 199
Part 178
sec 178.11
pg 391
PISTOL:
A weapon originaly designed, made, and intended to fire a projectile (bullet) from one or more barrels when held in one hand and having:
(a) A chamber(s) as an integral part(s) of, or permmanently alinged with, the bore(s);
and
(b) a short stock designed to be gripped by one hand and at an angle to and extending below the line of the bore(s).
RIFLE:
A weapon designed or redesigned, made or remade, and intended to be fired from the shoulder, and designed or redesigned to use the energy of the explosive in a fixed metalic cartridge to fire only a single projectile through a rifled bore for each pull of the trigger.
SHORT BARRELED RIFLE:
A rifle having one or more barrels less than 16 inches in length, and any weapon made from a rifle, wether by alteration, modification, or otherwise, if such a weapon, as modified, has an overall length of less than 26 inches.
Part 179
Sect 179.11
ANY OTHER WEAPON:
Any weapon or device capable of being concealed on the person from which a shot can be discharged through the energy of an explosive, a pistol or revolver having a smooth bore designed or redesigned to fire a fixed shotgun shell, weapons with combination shotgun and rifle barrels 12 inches or more, less than 18 inches in length, from which only a single discharge can be made from either barrel without manual reloading, and shall include any such weapon which may be readily restored to fire.
Such term shall not include a pistol or revolver having a rifled bore, or rifled bores, or weapons designed, made, or intended to be fired from the shoulder and not capable of firing fixed ammunition.