Finding a Break in an Electronic Fence

Are you talking about one of those buried wire fences

No problem.

Just use a variation of the method the electric utility workers use to find breaks in the underground service.

You "power up" the loop and use your high impedance DVOM to look for voltage gradients. If you DVOM can't measure the frequency produced by the "electronic fence" then replace that with a door bell transformer for the duration of the test. Stick your two probes into the ground a few feed across and keep doing this along the suspected route of the wire. You should measure a voltage equivalent to several percent of the driving voltage when you get near the break. Then move the probes one at a time in a place that gives the maximum voltage. You will be right on top of the break. You might want to put a bulb in series with the transformer drive just in case the wires re-establish contact while you are messing about.