What to use? Miss/Mrs?

Do you know her well, even without knowing her last name? In that case, you could say "Dear Julia," in the letter, but that's not very common business convention. How would you address the envelope to get her letter to her. Is there someone you can call to find out her last name?

But if she is simply somone you were once introduced to, and you don't know here very well, then you should use "Dear Ma'am:" I have also seen letters which were typed up "formally" with the Mr. or Ms. or Ma'am (or Madam, your preference), but then the person used a pen to write "Dear Mark" over the formal typed greeting. This would be for people who are on the "warm regards" standard of correspondence.

Oh, it's e-mail? E-mail doesn't follow many of the standards of proper business letters. You could probably use "Dear Julia" in an e-mail.  Pay attention to how she responds, though. If she addresses you as "Ms. Smith" and signs her full name in response, then future correspondence, even by e-mail, should go to Ms. Last Name.

Dear Madam is used in business letters when the surname of the addressee is unknown.

Dear Ann is acceptable if she is your colleague or friend.

If you don't know the surname of the addressee you can also use: Dear Sir, Dear Sirs;
Gentlemen (AmE)

P.S.
Complimentary Close
Yours faithfully - (formal) if you don't know the addressee (use it if the Salutation is Dear Madam/Sir/Sirs)
Yours sincerely - (semi-formal) if you know the addressee
Yours truly/Very truly yours - (AmE)

Regards
Dominik

if you do know her prename as well as surname would it be better in a business case to write Ms. Julia Smith or just Ms. Smith?? And when is the dot neccessary and when not??

Hope somebody knows and can help ;-)