David Yelland, the paper's editor, was forced to make a second apology, following publication of an apology in last Thursday's Sun, before Buckingham Palace agreed to draw a line under the matter.
Lord Wakeham, chairman of the Press Complaints Commission (PCC), issued a stern statement warning that "such a mistake must not happen again".
He added: "The newspaper's apology [following publication of the picture] in no way excuses the grave error which was made, nor lessens the distress which it caused Miss Rhys-Jones.
"The decision to publish these pictures was reprehensible and such a mistake must not happen again. In particular, I want to make clear that Miss Rhys-Jones should enjoy the same rights to respect for her personal life ... as anyone else."
The complaint of "unjust invasion of privacy" made by the Queen's communications secretary, Simon Lewis, last Wednesday, was the main item at the PCC meeting yesterday.
After the meeting, Lord Wakeham explained that the royal family had agreed to a resolution of the matter following a further apology from The Sun.
"As far as the formal complaint is concerned, Buckingham Palace has indicated to me today that this statement, together with the newspaper's own statement later this afternoon, resolves the issues they raised with us.
"There is therefore no remaining complaint on which the PCC will need to formally adjudicate."
Mr Yelland's own statement accepted the PCC's conclusion and reiterated his apology formally, if somewhat defiantly.
He said: "I made it clear last Thursday that I deeply regret the publication of intrusive photographs of Sophie Rhys-Jones and the distress caused to her," he said.
Obviously weary of the intense backlash to last Wednesday's publication, he added: "There will be no more comment on this matter which I now consider to be closed."
The Palace said: "Prince Edward and Miss Rhys-Jones, having seen the statement from the Press Complaints Commission and the further apology from The Sun, have confirmed that they will not be pursuing a formal complaint against the newspaper."
The PCC spokesman, Luke Chauveau, insisted that readers should not feel the tabloid had been let off lightly.
"What do they want - blood? The public does not always appreciate that it is a very serious matter for an editor to make a public apology. The main thing is that the complainant is happy. Miss Rhys-Jones can now get married without this matter hanging over her."
Last night Lord Wakeham was insistent that self-regulation of newspapers continued to be a satisfactory and successful option.
"This regrettable episode - which I have to say no conceivable privacy law could ever have prevented - does not detract from that in any way."