The victim, based in Hong Kong, paid the money into Lee's wife's bank account after Lee agreed to arrange for modifications to be made to the cue.
The money was transferred but the player, 39, continued using the cue.
Lee pleaded guilty to fraud at Swindon Magistrates' Court and was fined £110 and ordered to repay the £1,600.
'Drop the charges'
The court heard that when there was no sign of the snooker cue Marco Fai Pak Shek made a report to the police.
Prosecutor Michelle Hewitt told the court Lee had promised to send the cue along with letters of authenticity from himself and cue manufacturer John Parris but when the police carried out an investigation he was still using the cue as his own personal property.
Lee, who lives in Trowbridge, Wiltshire, was then arrested.
Ms Hewitt said Lee emailed Mr Shek saying he was going to personally deliver the cue.
The email read: "If you want the cue, you need to drop the charges and I will sign some pictures for you."
Mr Shek refused to drop the charges and the case continued.
Match-fixing
Mark Glendenning, for Lee, said his client had repeatedly apologised for failing to send the cue and had previously sold items to fans on his Facebook site without incident.
He said Lee had found himself embroiled in a number of difficulties which had taken up his time but "accepts in full he should have sent this cue".
Last month Lee's appeal against a 12-year ban for match-fixing was dismissed.
He was found guilty of seven charges in 2008 and 2009 including one concerning a World Championship match after a tribunal hearing in September last year.
He had also appealed against having to pay £40,000 costs, which have now been increased to £75,000.
The World Professional Billiards and Snooker Association said he was involved in "the worst case of snooker corruption we've seen".
Lee has been a professional player for more than 20 years and has won five ranking titles.

A new survey shows an overall increase of 11% in dog mess on UK beaches - it's gone up 71% in Scotland.
Dog poo is dangerous because it contains high levels of bacteria - if it gets into water it can be very bad for bathers' health.
Almost 4,500 volunteers took part in the survey by The Marine Conservation Society - across 335 UK beaches.
The society says that pet owners have been thinking ahead by carrying plastic bags to clear up dog mess - which is a good thing.
But it now wants to encourage them to take the bags off the beach and bin it in one of the many dog bins provided.
Even though levels of dog poo were up, there was some good news: overall litter levels were down this year.

Last week, Paul Kenny was granted bail and banned from attending games after admitting racially abusing Sinclair.
"It was disappointing," said Gers goalkeeper Foderingham, an ambassador for equality group Kick It Out.
"No-one wants to see that in football, especially at a club I represent."
Commenting on the incident, the 26-year-old Englishman said it was "difficult when one idiot comes and starts to behave like that".
"The club has dealt with it accordingly," he said.
"The Rangers fans have been fantastic. All season and since I joined the club, they have been superb.
"I haven't (spoken to Sinclair), I can understand that he is probably upset and frustrated, but hopefully he will be all right."
Meanwhile, Foderingham admitted the Rangers players are still getting used to new manager Pedro Caixinha's methods since he took over from Mark Warburton.
"It is a different style of play," he said. "He is an intensive, aggressive type of manager who wants his sides to play as such and it will take time for the boys to adjust.
"We are doing that at the moment and hopefully we can get a good pre-season under our belts and have a good go at it next season.
"Until we start winning football matches on a regular basis, the question marks are not going to go away.
"So what we need to do is work together as a squad, as a management, as a club and get back to winning football matches."
The former world number five snooker player Stephen Lee has been fined for selling his personal cue to a Facebook fan for £1,600 but failing to send it.
[NEXT_CONCEPT]
Pet owners who bag up dog poo and leave it on beaches are threatening the safety of people who visit the seaside.
[NEXT_CONCEPT]
Wes Foderingham acknowledged the hurt "one idiot" caused Scott Sinclair but says the fan, who admitted making racial gestures at the Celtic player, did not represent the Rangers support.