On the lathe turned down 7.5mm (three turns of the handle) of 5/16" brass hexagon bar to ø6mm as 1200RPM. Cut a 1mm thread at 40RPM along the turned down section. Used the tailstock chuck to drill at 2500RPM a ø2mm hole 11.5mm deep. Then removed the workpiece sawed it down to ~13mm and remounted it by screwing the new threads into a holding piece mounted in the lathe's chuck. Used a centre drill then at 2500RPM extremely slowly fed a carbide ø0.3mm bit in with the tailstock. It is an amazing testament to the lathes quality that the run-out didn't snap the drill-bit. Faced at 1200RPM the piece down to a length of 12.5mm and then tapered back slightly. Then cleared the nozzle hole that was blocked by the facing.
The 0.3mm hole seems to grip the drill-bit so can't be noticeably oversized, looks dead centred round and strait. As always Richard's advice and higher-quality drill-bit were vital.