Once you've located the break you can easily make a bypass to keep the fence running.
Because the wire should only be buried just under the surface of the soil, it's easy to dig and rebury once it's fixed.
Once you've located the break, unplug the fence and carefully dig up the wire. You'll need enough uncovered that you can draw it up and work with it.
You will need a new piece of wire to splice into the old one. For a secure, waterproof connection you'll need wire nuts and wire tape or a shrink tube.
Clip the wire apart at the break, and cut out the broken length. Use wire strippers to remove about an inch of the wire insulation on each exposed end of the old length and the new piece. Twist the end of the old wire together with the new one until they are securely intertwined. If using a wire nut, put both ends into the nut and twist until it's secured. Wrap all of the exposed wire with high-quality electric tape. This makes the splice stronger and helps secure it even further. It can help the splice be resistant to rain as well. Use a shrink sleeve and heat gun to ensure the connection is completely waterproof if you'd like. These are easy to find at electrical supply stores.
Repeat steps 3-5 with the other end of the new wire.
Test the new connection by plugging the wire back into the terminal and testing with the collar to make sure it works. Rebury the wire.
If your yard is small and you don't want to mess with finding the break, you can opt to run a new wire after you've confirmed that the transmitter and collar are working fine.