Dear …
Please allow me to take this opportunity to apologize to you for the recent situation that occurred between you and __(a customer service rep/an employee/etc.)__ in our __(type of)__ department. After my investigation into this matter, I discovered a couple of things.
Firstly, the information that was passed along to you by this individual was correct, as __(this is company policy/these are the regulations we must follow/etc.)__. I further discovered, however, that the way in which this information was delivered to you, lacked __(professionalism/empathy to your situation/etc.)__. And for this, I sincerely apologize to you on behalf of __(Name of Organization)__.
It is not the intent of our organization to appear __(indifferent/uncaring/etc.)__ towards our customers. Quite the contrary actually. We do concern ourselves with the individual circumstances of our customers. With this in mind, please allow me to advise you that the individual involved has been spoken to about the mannerism in which __(he/she)__ came across to you.
Again, we apologize for that incident and genuinely hope that there will not be a reoccurrence of this same type of situation again.
Sincerely,