Would a 1N4004 diode between the amplifier and the analog pin solve the problem?
Maybe... but, the diode won't start to conduct (in the "positive" direction) until it has about 1/2 Volt across it. That might be OK If you are only trying to capture loud sounds, and your signal is strong enough. But, quiet sounds below 1/2 volt will be cut-off.
A better solution is to wire the diode "backwards" to ground, so that negative voltage turns the diode on, and the diode 'shorts out" the current. (You'll still get about -1/2 volt into the Arduino, but that's OK.) You will need a current limiting resistor, so that your op-amp doesn't get damaged with the negative current "shorted" to ground But, since you should already be using a voltage divider, those resistors will do the job.