Mr Neilson successfully shot one chamois and two tahr.
However, he'd failed to obtain a hunting licence, despite them being available online and free of charge.
On the evening of March 9, Neilson was sighting his rifle at a nearby creek when a native whio landed nearby.
Neilson took aim at the whio and shot it through the chest, killing it instantly, Christchurch District Court heard today.
Whio, which appears on the $10 bank note, is classified as a nationally vulnerable species, with current populations declining, especially in the South Island. There are between 2500 and 3000 left across the country.
DoC says that unless the causes for its decline are remedied, or reduced, the species faces a "very real risk of becoming extinct".
Two other hunters who were in the area heard the disturbance and investigated.
They informed Neilson that whio are a protected species and later complained to the Department of Conservation (DoC).
Later, when interviewed by DoC officers, he explained that he shot the whio "because I was a big idiot, brain dead".
Neilson appeared at Christchurch District Court today to plead guilty to charges of hunting and killing protected wildlife, and using a hunting weapon without a permit.
Defence counsel Richard Maze said the incident was a mere accident, borne out of a lack of research and "one moment of very poor judgement".
He grew up hunting, has been a licensed hunter since he was 16, and has been a member of hunting clubs.
Neilson was a responsible hunter, Mr Maze said, and had done his research before arriving in New Zealand to hunt chamois and tahr, but it had been "inadequate".
He thought the duck he shot was a common mallard, and not an endangered whio, of which he "wasn't even aware of their existence", Mr Maze said.
He asked for Neilson to be sentenced today so that he could fly back to Denmark on Tuesday.
DoC prosecutor Susan Newell stressed the importance of hunters identifying their targets before shooting.
The whio is taonga to the local iwi, which is Ngai Tahu in this instance.
Judge Brian Callaghan fined Neilson $7000 and ordered him to make a $3000 contribution to DoC's costs in bringing the prosecution.