Manchester City missed out on a Europa League place after ending their miserable Champions League campaign with a defeat by Borussia Dortmund.
Ajax's defeat by Real Madrid meant City would have finished third in Group D had they beaten the German champions.
But Dortmund posed the greater threat, forcing a series of saves from Joe Hart before Julian Schieber slid in to score from Jakub Blaszczykowski's cross.
City crash out of Europe without a win in any of their six group games.
Their haul of three points is a meagre return given the vast resources at Roberto Mancini's disposal, and is the lowest by an English club in the 20-year history of the competition.
City crash out with new low
Manchester City's tally of three points from their six group games replaces Blackburn's record in 1995-96 as the worst by an English team in the Champions League. Rovers finished bottom of a group containing Spartak Moscow, Legia Warsaw and Rosenborg on four points, with their only victory coming in their final game against Rosenborg when Mike Newell scored a nine-minute hat-trick.
City become the first Premier League side not to manage a single group win, and missing out on the consolation prize of the Europa League means their seeding is likely to drop further when they return to Europe next season.
On this evidence, a vast improvement is required if they are to make any impact. If it had not been for some superb saves by Hart, City would have lost by an even greater margin to a depleted Dortmund side without many of their stars.
Jurgen Klopp's side were already through as group winners so the onus was on the visitors to chase victory, especially with Ajax behind early on in Madrid.
But Mancini's men created little in the first 45 minutes, and their only effort on goal was a bouncing shot by Edin Dzeko that Roman Weidenfeller pushed round his post.
Carlos Tevez worked hard and covered a lot of ground but, without the injured David Silva, City were lacking a spark in attack.
Dortmund also began slowly before creating a host of chances either side of half-time.
Hart, City's hero when they snatched a point against Dortmund at the Etihad Stadium in October, came to their rescue before the break when he got his fingertips to a Marco Reus drive and dived to keep out Ivan Perisic's curling shot.
Mancini unfazed by European exit
The home side went closer still when Mats Hummels was just unable to turn in an inviting free-kick from Blaszczykowski, who came on for Marco Reus at the interval.
Edin Dzeko headed straight at Weidenfeller as City tried to respond but the chances kept coming for Dortmund, with Hart tipping a swerving Perisic volley over the bar and reacting well to beat away Kevin Grosskreutz's fierce strike.
So it was no surprise when they took the lead just before the hour mark, Schieber gratefully gobbling up Blaszczykowski's delightful low cross.
The scorer almost added a second soon after when Hart kept out his chip and City's response was muted at best, despite the best efforts of Tevez.
The Argentine striker and his compatriot Sergio Aguero forced smart saves from Weidenfeller but Hart remained the busier goalkeeper, stopping a Robert Lewandowski effort with his face.
In the end City went out meekly. They had known before kick-off they could progress no further in the Champions League and this lethargic performance was an appropriate way to end a hugely disappointing campaign.
A far better display will be needed when the Premier League champions return to domestic action on Sunday in what promises to be a pivotal clash with title rivals Manchester United.
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