Labour is down three points to 31 per cent and Mr Cunliffe is down 2 points to 8 per cent.
It has National on 47 per cent, down from 51 per cent last month.
The Herald DigiPoll survey published in early March had National on 50.8 per cent and Labour dipping into the dreaded twenties, on 29.5 per cent.
The big shift in the One News poll is a jump in support for Winston Peters' NZ First party, up by four points to 7 per cent, and well clear of the five per cent threshold required to get seats in the House without an electorate seat.
Polling for both the TV3 and One News polls finished last Wednesday, the day before the Internet Party launched its membership drive, but in the midst of publicity about a potential alliance between the Internet Party and Hone Harawira's Mana Party.
The Herald on Sunday revealed a week ago that Mr Harawira and Mr Dotcom, an internet mogul, had held talks with a view to forming an alliance ahead of the September 20 election.
The Mana Party will thrash out the issue at its annual conference on April 12 and 13.
The Internet Party is yet to be registered or reveal who its leader is or name any candidates.
Mr Dotcom, who is wanted in the United States on criminal charges of copyright, money laundering, racketeering and wire fraud, is not eligible to stand because he is not a New Zealand citizen. But according to the party constitution, he is the "party visionary."
TV3 Reid Research Poll:
National 45.9, up 1.4
Labour 31.2, down 2.3
Greens 11.2 per cent, down 1.2
NZ First 4.9 per cent, down 0.8
Conservatives 1.9 per cent, down 0.2
Maori Party 1.5 per cent, up 0.3
Act 1.1 per cent, up 1.1
Mana 1.1 per cent, up 0.8
United Future 0.1 per cent, up 0.1
Internet Party 0.4, up 0.4
One News Colmar Brunton Poll:
National 47 per cent, down four points
Labour 31 per cent, down three points
Greens 11, up three point
NZ First 7 per cent, up four points
Conservatives 2 per cent, up one point
Maori Party 1 per cent, no change.