Please accept my sincerest apologies.

Meaning

I am truly sorry.

Examples

• John: 'I'm so sorry I missed your birthday.'
• Alice: 'Please accept my sincerest apologies. I had a prior commitment I couldn't reschedule.'
• John: 'Please accept my sincerest apologies. I should have communicated more clearly.'
• Alice: 'I'm truly sorry for the misunderstanding.'