Step 1: Base
Once I got it home I completely disassembled it (I didn't take a photo of it before I took it apart, but there is one of the box!) I had to cut the wires from the socket, because I couldn't undo them, then I could remove the reflector, the corrugated plastic tube on the arm, and the nuts fixing the arm to the base. I used a hack-saw to remove the upper gooseneck half of the arm, because it wouldn't help support the bamboo pole, then re-attached the arm to the base and ran the wires back up inside.
I then cut the plastic tube in half: one half I slit open and glued it around the metal arm to make it a bit thicker, the other I slid inside the bottom end of the bamboo pole (I was quite lucky that the diameters of the lamp bits and of my bamboo were fairly close!) I also re-used the plastic surround that was originally at the top of the gooseneck arm under the reflector, gluing it upside-down to the base to sit the bottom of the bamboo on. In the third photo you can see the base with the arm reattached, and the plastic tube glued around it, in the fourth you can see the gooseneck bit of the arm that I sawed off, and the reflector. My original plan was to build the reflector into the base for more weight, but this didn't work out (especially once I'd mangled it with a scrappy attempt to cut a hole through the metal).
Of course, the little 12-cm diameter base doesn't stand a chance of supporting a 1.3m pole, so I cut a length of bamboo, split it lengthwise and screwed these feet onto the base (it contains a thick rubber "puck", which was easy to screw into), to make the base more stable. Pre-drill holes through the bamboo for the screws or else it will split - they can be slightly narrower than the screws so there is some bamboo left to bite into, though. Drill from the top (the outside), because the fibres will tear when the drill punches through, and it's neater if this is on the bottom - they can peel along quite a way.
Then it was simply a case of soldering on an extra length of cable to run up inside the bamboo (heat-shrink tubing to cover the joins), then pushing the cable through the bamboo and fitting the pole onto the arm on the base (the plastic tubing inside the pole fit snugly around the plastic tubing I had glued around the metal arm, holding them quite firmly together).
The lamp socket from the desk lamp already had the angle bracket attached to it, so I just had to drill two small holes in the top of the bamboo pole and screw the bracket on. A scrap of bamboo quickly cut to size and slid into the top of the pole protects the cable from the ends of the screws on the inside. Wire the socket to the cable - I soldered the wires and used heat-shrink tubing, for neatness, and then used hot glue to keep the wires running together down the back of the pole, so they stay hidden - screw in a CFL bulb and the electrical bit of the project is finished: if you like your lights industrial you can leave it at that ;) I thought I'd add a shade to mine...