During certain major life events, such as a marriage, divorce, or adoption, a name change may occur and you'll need to notify various parties of this so that they can update their records.
If the letter is being sent as a hard copy, then it should be in the proper business format, otherwise, if the letter is being emailed, it should have an appropriate subject line and proper grammar.
I am writing to inform you that I've recently gotten married and have changed my name. My name was Meredith Smith but as of June 25 my legal name has changed to Meredith Jones. For your reference, my member number is A18543. I have enclosed a copy of my marriage certificate as proof of my name change.
If you have any questions, please contact me at 555-555-5555.