la main sur le cœur

Etymology
Putting one's hand on one's heart when talking is seen as a gesture of sincerity.

Adverb

 * 1) in all sincerity, open-heartedly, wearing one's heart on one's sleeve

Usage notes

 * Not to be confused with.
 * Generally carries a negative connotation; used when someone is duplicitous enough to lie while giving the appearance of telling the truth and only the truth; compare "to lie under oath".