The correct phrase is "an ear." The article "an" is used before words that begin with a vowel sound, while "a" is used before words that begin with a consonant sound.
In the case of "ear," the pronunciation begins with a vowel sound (/i:/ sound). Therefore, you would say "an ear."
"She whispered into his ear."
Remember, it's the sound that matters, not the actual letter. So, even though "ear" starts with the letter E, which is a vowel, it is the sound that determines the correct article to use.