President Mwai Kibaki is staking his political prestige on the referendum which has turned into a power struggle with opponents.
They Mr Kibaki has failed to end decades of graft and tribalism in east Africa's most developed nation.
The tussle between Mr Kibaki's "Yes" camp and his opponents' "No"
campaign has widened a split in his cabinet.
It’s also deepened the strain of communal tension in the country of 32 million, seen by Western allies as a linchpin of stability in a turbulent region.
The main controversy over the charter centres on the powers of the president.
Critics say it ignores the desire of most Kenyans to balance those powers with a strong prime minister's post and other checks.
Police deployed
Paramilitary and regular police units have been deployed overnight at what they called potential flashpoints in the coffee-growing country.
Political violence during the campaign has left nine people dead and dozens wounded.
Witnesses reported scores of paramilitary officers armed with assault rifles on the edge of Nairobi's volatile Kibera slum, an opposition stronghold, yesterday evening.
"We have asked for more security in certain areas which are considered flashpoint areas like Kibera," Mani Lemayian, a spokesman for the electoral commission, told Reuters.
"This is normal practice in areas where there may be tension, but no one should feel intimidated to vote."
Mirugi Kariuki, assistant minister of internal security, told Reuters all police leave had been cancelled for the vote, the results of which are due early on Tuesday.
Analysts said the arrival of the feared paramilitary General Service Unit in Kibera might stoke tensions there, noting there was no such deployment during peaceful elections in 2002.
"That is clearly intimidation," said Koki Muli, a poll expert organising a 19,000-strong force of Kenyan observers during the referendum.
"It just does not help. Everyone is on edge, scared and nervous as it is."
Sociologist Ken Ouko added: "The signal sent by the government is that it is in fear.
"It should wait and at the first sign of trouble deploy. Doing it now only raises the temperatures."
Kibaki appeal
President Kibaki urged Kenyans to avoid clashes during the voting, warning that unrest might deepen divisions among Kenyans.
"Let us reject violence and selfishness which could destroy our country's unity," Mr Kibaki said yesterday in a speech.
"It is our duty to ensure we conduct this vote in peace.”
Riots engulfed Nairobi for three days in July when the constitution was published and at least one person was killed.
A further eight have been killed in riots around campaign rallies.
Mr Kibaki, 74, says the new charter will improve governance in a country ruined by decades of "Big Man" rule and theft of state funds under his strongman predecessor Daniel arap Moi.
In reality, critics argue, Mr Kibaki is staging the vote to try to quash cabinet dissidents who accuse him of concentrating power around his Kikuyu tribe and permitting a group of business cronies to loot state coffers, a charge he denies.