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.

The Islamic Center of Boca Raton had planned to host a polling site for the state's primary in August and the general election in November.
Officials rescinded the invitation, drawing sharp criticism from Florida lawmakers who said it reinforced religious discrimination.
The site was moved to a nearby library.
County Elections Supervisor Susan Bucher said she moved the site after receiving about 50 complaints from people who said they did not want to vote in a mosque.
The Islamic Center has been used as a polling station at least since 2010, the Washington Post reported.
Democratic US Representatives Ted Deutch and Lois Frankel released statements opposing the move.
"If we are going to use places of worship as polling places, we should not discriminate,'' Mr Deutch said.
It is unclear how many houses of worship are used as polling places across the country, but churches are often selected as host sites because of their large auditoriums and parking lots.
Mosques in California, Iowa, Pennsylvania and Ohio have served as polling places, including one that has been used since at least 2004, according to the AP news agency.
Norway has rejected a plan to give its neighbour Finland a mountain to mark the centenary of its independence.
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A Florida mosque has been removed as a polling station for the 2016 election after local officials received complaints and threats of violence.