Active Listening is the best skill I believe that could help you utilize a conflict, by active listening you can gain a better understanding from where the other party is coming from.
Using silence while they explain their side of the story and not interrupting them can be helpful so that they feel that they have been heard.
It is important to remove any ill feelings or personal judgments, These may stop you from meeting a equal conclusion. Prior involvements with this person may affect how you handle this conflict, maybe feeling like they ‘deserve’ to loose.
I would also implement the use of an open body language to show the party that you aren’t already against what they are about to say.
Using reflecting skills may also be helpful, repeating what they said and rephrasing your interpretation back to them may help both parties understand correctly, removing some of the chance of misunderstandings.
From here in my opinion I would make a clear statement of what I want to occur but be clear that it is open to compromise, which respects the other persons feelings. What is important at the resolution of a conflict is that both parties feel like they have achieved something and don’t feel like they have come out the looser with bad feelings towards the other.