The manufacture of seamless tubes in which the process includes providing an assembly having a metal tube blank, and an elongate metal core of shape memory effect material which is surrounded and contacted by the tube blank with a minimal gap. The assembly is elongated by mechanical working thereof at an elevated temperature until the tube blank has been converted into a tube of desired dimensions. After the elongation step, the core is subjected to a treatment which results in the core being in a stretched condition throughout its length, and does not substantially stretch the tube. The core is removed from the tube, and subsequently subjected to drawing passes over a nondeformable mandrel thereby refining the precision of diametric and wall dimensions with improved ID and OD surface quality. There is also decoring and reinserting to improve final dimensions which results in the ability to fabricate smaller, longer tubes.