Hi and welcome at CDfreaks!
external drives (at least the normal half-height ones) are normal internal drives put into external enclosures. So you could install that drive internally without the need of additional cables (the latter depends on the configuration of you computer). Also you should be able to put an existing internal drive into the external case.
The main difficulty may be the external case, some are screwless, so there is a risk of damage.
P.S.: What drive is built into your new system? It is hard to believe, that the drive is not capable of writing DL discs, unless totally outdated hardware is used.