I have had to do this on several occasions.  Because the person was not performing well and also because of general workforce reductions.  I have also had to let go of a spouse while the other spouse still worked at the company. 

Letting someone go under is hard under any circumstance, but it becomes even harder if it is someone you are close to.  There have been studies that suggest managers can suffer from PTSD from stressful firing situations. 

The question asked specifically about how to cope as a manager vs. how to actually terminate the individual, so I will focus on that. 

1. Make sure you personally terminate the employee.  You will often have a representative in the room with you, but you should be the person to drive the discussion.  Yes, this is harder for you in the moment, but you will actually feel better about it later on.  The person you are terminating will also appreciate it (maybe not in the moment, but after the fact they will).  It shows a level of respect for them and demonstrates that you are willing to bear some of the burden.
2. Do it as soon as you know you it has to be done.  Fretting over it will cause you so much anxiety that you will not be able to focus on anything else and it could also affect your health.  I know that I have personally lost sleep and been so anxious my hands would shake before I had to go through with some terminations.  The hardest part for you will be leading up to the actual termination.  Yes, it will still feel bad afterwards, but nothing like the anguish waiting to do it.
3. Say the absolute minimum during the termination.  When you are nervous, you will be inclined to keep talking.  Saying too much can actually lead to legal issues, but it is not really doing anything to make the person feel better.  It is an attempt to justify why you are doing what you have to do.  Show them respect, say what you have to and promise to keep in touch.
4. Follow-up with the person shortly after they are terminated.  Give them a day or two to personally deal with the termination and then reach out to them to offer your support and reassurance.
5. DO NOT participate in bashing your employer.  You can listen to them vent all they want, but do not take part in being negative about your current employer. 
6. Work with a coach.  If you can find a mentor or a coach, ask their advice for dealing with situations like this.
7. Sometimes having a professional to talk to (like a therapist) can help you deal with any guilt that you may have.
8. Do something physical to help get the tension out.  Yoga, Gym, Boxing, etc.
9. If you feel strongly that this person  wronged then start to look for a new job yourself.  Do not compromise your morals. 

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