That's going to be tough. You can try Izotope RX restoration suite. It's not too expensive and has some good tools in there.

My honest opinion is that you just move on.
The sound clip link you have provided doesnt really tell us much.

Does the vocal you want to remove play simultaneous with the one that you want to keep? If not just edit out in btwn the good vox. If it does live together, than your options are few and poor.

1-Try eq ing out the unwanted vox. But you will most likely effect the vox you want to keep as well, therefore this is unlikely a good option

2-Try waves brand plug in, X-noise Removal. How ever these kind of plugins work with frequency analyzation techniques that need a part of the clip that has the bad audio alone in order to "learn" what to take out. (in most instances) Again this kind of plugin may alter the frequencies too much and may not leave you with what you are looking for.

3-Then you have the Cedar plugins like the NR-4 and others. Great plugins if you are rich.

IMO make rerecording it right an option and get a good quality recording.

The audio as a whole is very important to me, and re-recording is not an option. I am willing to go to whatever lengths neccessary in order to improve the audibility of the second voice.

Ahhh!
Desperate times call for desperate measures!

If you're looking for plugins then I assume you have a DAW capable of lowering the volume of parts within an entire track.

I have Vegas. What I would do in this instance is hit 'V' which adds a volume edit line in the track. Then find the parts where your voice is and add points before and after, and drop the volume between those points.
Next, I'd add range markers at each in / out instance of your voice and the other speaker's - the offending noise will be more prevalent between those markers. Next, raise the volume of the entire track via the fader. Once you're there, insert a noise removal plug and try to find a setting that will get rid of the background noise. If you find a setting that works, add a second track and move each instance of your voice (between the markers) down to it.
Apply the noise removal plug to the other party's voice on the original track.

If that's a no go, you may get lucky finding the more annoying HF and LF noise with an EQ.

If that doesn't work, and this track really is that important to you, seek professional help - someone with Cedar as mentioned earlier might be able to do what you cannot.

Edit> Having listened to the sample you posted, I don't think you'll ever get this to a point where it could be useful for anything outside of a reference for you personally... I can hear another voice in there, but man is it buried.

Yeh, I would try going back & forth between Waves X-Noise & or Z-Noise & turning up the volume with L2, or L3 Ultramaximer many times. Maybe do it in Wavelab so you can continuously render & see the waveform, and of course also EQ. I think that's going to be the three tools that will help. I think Bias Soundsoap might work good there too. You can tune in & out the bad noise with that.