Firstly, and very obviously, don't remove the old wiring until you have its replacement ready, because you'll need it to pull the new wiring through.

Secondly, while you may get away with taping the new to the old, for a stronger and lower profile temporary join, clean the black off the ends of the old wire and solder it to the bared ends of the new. (Edit: Er, and don't do what I did and forget to disconnect the affected wiring from anything live - I managed to blow a relay - replacement cost Ł12,00)

Edit: note that the blackening may extend a long way along your old wire - I have had to replace entire runs of internal wiring on Freestyle wherever there has been a junction below floor level. Presumably at some stage the junctions have been immersed in water because it has clearly tracked along under the insulation, uphill (by capillary action). I am now such a devotee of soldered junctions protected by heat shrink tubing that Freestyle never goes anywhere without her 12v soldering iron.