I'd just use JB Weld, or some other similar epoxy. Should be more than strong enough, and because it still has that little rebate, it should locate itself back together without much stress.

with the cable through the middle, any repair requiring heat is out of the question. Maybe something like JB Weld or other Epoxy based metal bonding solution.

Supercheap Auto $20

Bunnings $18

Thanks for the feedback so far guys. What (clamping) tool do you recommend I use to hold it together while it sets? 

Bung: I don't fancy the chances of just epoxy being strong enough. I had a similar lamp where the join was a short threaded tube that screwed into both upper and lower tubes. When that broke I didn't have a replacement but was able to shove a length of alloy tubing salvaged from an old tv aerial into both sides. I would try to find some tubing that would fit inside the stand while allowing the cable to fit to reinforce the join.

i agree with the epoxy probably not being strong enough on its own. An insert would be good but if using a tube, you’d need to remove the cable, glue the insert in, then re-thread the cable. If you can remove the power plug and pull the cable up out of the lower half of the stand tube, this method may be not too much of a fiddle. Use the ‘5+5 cm aluminium tube’ idea below.

However if removing the cable is not possible, maybe you could find some aluminium tube at Bunnings or M10 - with external diameter the same as (or even very slightly smaller than) the internal diameter of the lamp tube. Cut a length of say 10 cm then cut that in half lengthwise.

The idea is to make two internal splints. Glue each half into the lower part of the lamp tube to depth of 5 cm (i.e.5 cm remaining exposed) with the cable remaining in place in the centre of two aluminium halves, and let the epoxy cure overnight. Then glue the top part of the tube down over the exposed splints.

You probably wouldn’t need to clamp this at all - if it’s a snug fit it should be self-supporting while second lot of epoxy cures.

This version depends on whether you can get enough slack in the cable exposed at the break to get the two parts of the lamp tube far enough apart to get the insert pieces in place. You might be able to release the cable at base of the lamp.

Thought of a possibly much easier way. Epoxy the two halves back together. Get aluminium tube with internal diameter same as the external diameter of lamp tube.

Cut a 10 cm length and cut lengthwise into two splints and epoxy them on the outside of the join to reinforce the join. Temporarily hold the splints in place with hose clamps while the epoxy cures.

It looks like there’s two different external diameters at the join. Work with the larger one - and where it’s narrower, let epoxy fill the gap.

Mask tape and spray paint the splints black to match - or buy a black test pot from Resene and paint by hand.

Thank you all for the advice so far. I'll see what I can do this weekend and post the results. Though, most likely won't be done until next week. There's definitely enough cable slack to insert something into the pipe.