I have several students who are singers. When they sing in front of an audience, generally no one knows that they are not American. Then, when they talk between the songs, people are shocked to hear their accent. This is common.
Here is a very interesting podcast discussion on this subject by several linguists. I think you will enjoy it.
http://www.waywordradio.org/do-singers-have-accents-minicast/
I love working with singers because they almost intuitively know how to produce the new sounds that I introduce to them. They understand how a slight change in tongue position can alter a vowel sound and they can easily understand the difference between voiced and voiceless consonants.
If you have no talent for singing, it does not mean that you will not be able to change your accent. Many people who cannot sing well also learn very quickly. Motivation and regular practice are the two most essential elements to success.