First, keep calm. Then, consider these points:
1. Control your emotions: Chances are that emotion is at an all-time high so your natural instinct is fight or flight. But that will only make things worse. Fighting would be like adding fuel to a burning fire. Withdrawing would make the other party more frustrated at the situation--forcing them to act on emotion rather than logic. Instead, apologize as your opening response; this is a tool for diffusing the emotion and for keeping all parties sane.
2. Establish rapport: Clarify what misconception or misunderstanding your customer may have about your role as anything other than an advocate for them.
3. Express understanding: While it may be impossible to predict the future, provide a plan on what you will do to help mitigate surprises. Even though this time your only option may be to offer an apology, it still signals that someone cares.
4. Ensure success: Promise what you will do and do what you promise. Nothing reassures your customer more than seeing for herself that you follow through on your plan--even if this means lots of caffeine, late nights and weekends.