Just to explain Jim's suggestion in a little more detail
Strip the covering from both wire ends .
Twist the strands of wire at the end of each cable (but don't join them together yet.)
Then solder each cable end seperately with a thin coat of solder .Put the solder iron tip firmly on the one of the bare wires and heat the cable until the solder melts instantly when its is touched onto the wire and runs along the twisted wires. Repeat for the other cable .This is tinning.
Then put a good length of heat shrink over one piece of the cables you are joining and lay the tinned cable ends together so that they are in a line and solder them. Don't move them until they have cooled or you'll get a 'cold joint' .
You need to hold the soldering iron tip on the two tinned cable ends at the same time and wait until the solder tinning melts then add solder to the tinned ends so that it flows quickly and instantly and joinss both cables together. If it the solder doesn't flow instantly you haven't got the cable ends hot enough
Then pull the heat shrink over the joint and run the blade of the solder iron over the heat shrink until it shrinks tightly onto the cable.
