。
I'm really, really sorry.
Literal
Truly pardon-please [softener-ね].
ごめんなさい is the polite/plain apology derived from 御免 ('pardon / dismissal') + なさい (the gentle imperative ending, here fossilized). Softer and more personal than the highly polite すみません, often used by children or in feminine speech. Final ね adds a coaxing, almost pleading tone.