What do I do about three employees who seem to have constant interpersonal conflict?

It's all about communication. First, invite the other manager for a cup of coffee (or whatever), and lay out the problem as you see it. Listen to her response, she may have helpful insight.

Suggest to her you would like to provide a "third-party intervention" and explain you would like to have them in a room with certain guidelines (rules) agreed to before you start. Then, one by one, let each say what is on her mind. One of the rules is, no interrupting someone who is speaking.

Use a white board to document each person's grievance or view of the situation in the form of an issue/problem. Seeing their "situations" in writing will give them a different perspective on their own concerns, as well as the others.

Then, once each has had a say, ask each what she thinks the solution is. Again, use the white board to document their responses. Once they agree on a solution or solutions, draw up a contract for them and have them sign. Give each a copy. Typically this will work.

If nothing else, they will work harder at getting along because they don't want to go through that process again.

I agree with both responses 1 and 2. I have utilized both practices depending on the situation and communication style of those involved. It definitely is all about communication and problem solving. I have found when co workers lack the skill of effective communication and problem solving, it is important to go back to the rule book, the employee handbook. We have implemented a ZERO Tolerance Policy which includes creating a hostile work environment. This policy is taught and practiced throughout the company and it empowers all managers to ensure that we maintain a safe and healthy work environment at all times. This policy gives all managers the responsibility and authority to address matters that affect the work environment; regardless of who is the employee’s manager and covers the behavior of all levels of management.

I would then speak to each person involved reminding them of the policy and how the behavior is violating the right of other employees to a healthy, safe and welcoming work environment. Then give them the choice of how they would like to resolve the matter, and then mediate. I always bring them back to the basics.

The first step is getting a commitment from each employee that they want to work it out. The goal is not for them to become best friends but demonstrate to them that their bickering is interfering with business performance...and that's intolerable. Any time you and they are wasting on resolving the conflict is missed opportunity.
If you don't get a commitment, then you have one solution. But from my experience, most employees will at least express interest in working it out. A next step is using a tool like CriteriaOne DISC and Business Values and Motivators. In the majority of cases, the conflicts are really just misunderstanding in the unintential way people approach one another. The Golden Rule is part of the problem, not the solution.... it doesn't work to treat people like YOU want to be treated. It's more important to treat people like THEY want to be treated. That's called the Platinum Rule. Using DISC and BVM is an inexpensive next step to help people understand their preferred styles of communication and then more importantly how their approach is perceived by others. For instance, one person's style might be just to say it like it is. For people like them, candor might be valued. But for other, directness might be seen as being bossy. Another example is detail-orientation. One might see dotting every "i" and crossing every "t" as doing things right; others see it as being nitpicky. Perception is reality. When individuals realize that their approach to problems, people, pace, and procedures isn't a personal attack on other but their personal preferences, more than half of conflicts are resolved. I've even seen quite a few former enemies become friends. Then that leaves the few people who continue to get on each other's nerves. For those conflicts, I'd suggest following the good advice of the other experts.

Philip:
I think the other answers are great. I would introduce another dimension to consider as well. I have found that conflict usually comes from one of two core areas:
Identity Conflict- identity conflict speaks to my personal values and who I am. If I feel disrespected or unvalued intellectual solutions don't work. This is visceral what Seth Godin calls the "lizzard brain"
Interest Based Conflict- this is where we have or think we have competing interests and goals. This is a great opportunity to building bridges and asking them to construct best case scenarios so everyone gets what they "need".
You have another issue- the other manager. You two aren't in synch. You both need to be on board and this may require intervention/support from your mutual boss.
I am actually working an organizational intervention like you describe right now. I refused to take it on unless the managers agreed to participate.
The other thing is that if you have a true identity based conflict based on values you may have a bad "fit" and those are typically unfixable. Unfortunately when we hire we often stop at the surface and that leads to problems down the road.
Another intervention is a little more abrupt. You do the problems solving and then go to zero tolerance- you tell the three that continued disruption will lead to corrective action up to and including termination. We aren't a club we are a business. Scream at each other on your own time.....

Great question Philip on a situation that happens in many workplaces. I agree with the other experts that communication is key. It's beneficial to set up an environment where people feel comfortable talking with each other and can do so on a deeper level. So much conflict occurs because of misunderstandings as well as what is going on inside us.

When we set up a conversation where people can talk with each other openly, calmly and without rebuttals it opens the door to people understanding each other and building empathy. The idea is to get rid of the conflict by going to the root of the issue and letting people share their stories in a safe environment that encourages positive behaviors. http://www.guyfarmer.com/conflictresolution

do team building counselling and put them all on team work training or joint project, needed to be completed by three of them. make anyone as leader and tell him the qualities of leader, and the rest of two can be followerers and should be told about how to liten to boss and advatages at time of performance appriaisal, promotions, increments etc. hope they will be fine. if not change one of them to other department under strong manager- break the glass.