No one wants to admit weakness in an interview. However, if you are able to frame your response properly, you can show off more of your strengths than weaknesses.
"I sent an email at work without proofing it and then later realized that there were two people copied who were not suppose to know the information I shared. As soon as I noticed, I spoke to my boss and asked what I needed to do to resolve the situation. At that point, all I could do was apologize and learn from my mistake. Ever since then I am extremely careful to double check everything, whether it is an email, report or the status of a patient in critical care."