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England needed to beat France by 26 points to deny Ireland the title but came up six points short at Twickenham.
They finished second on points difference behind Joe Schmidt's side.
"Credit to Ireland, they beat us away from home, so congratulations to Joe and his team," added Lancaster.
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"We're sat here disappointed but we've just put 55 points on France and scored seven tries in the process.
"I'm hugely proud of what that England team has shown today and throughout the Championship."
This year was Lancaster's fourth tournament as head coach and England have finished runners-up on each occasion.
England's victory over France was a fitting climax to a tournament which concluded on an extraordinary, record-breaking day.
Wales won 61-20 against Italy in Rome before Ireland beat Scotland 40-10 at Murrayfield to go to the top of the standings.
Only once had England beaten France by the required amount - the 2001 Six Nations encounter at Twickenham, when England won 48-19 - and Lancaster's men fell short again.
Head coach Stuart Lancaster has described England's 55-35 victory over France on the final day of the Six Nations as "one of the most courageous performances" he had ever seen.