A social media campaign had proposed giving Halti mountain summit to Finland for its 100th birthday next year.
The border between the two countries runs up the mountain near its peak.
But Norwegian Prime Minister Erna Solberg said that sadly she had had to turn down the idea because the country's constitution prohibited any sacrifice of Norwegian territory.
Part of Halti mountain is already in Finland - and forms its current highest point - but the peak is 20m (66ft) across the border inside Norwegian territory.
Campaigners had proposed moving the border to give the summit, at 1,330m (4,363 ft), to Finland, saying the change would be barely visible on the map.
They had argued that the peak "would be a wonderful gift to our sister nation" to mark its independence from Russia, and not a big loss to mountainous Norway, which has much higher peaks.
The campaign's Facebook page got 17,000 likes, with support from both sides of the border.
But it hit upon an insurmountable legal issue.
"Border adjustments between countries raise challenging legal problems, among them linked to the Norwegian constitution," Ms Solberg wrote to Svein Leiros, the mayor of the town of Kaafjord in northern Norway, who supported the campaign.
Norway's 1814 constitution stipulates that the country is "indivisible" - and apparently not even an area the size of a football pitch can be split off.
"We will instead consider another suitable gift to Finland on its anniversary," the prime minister added.