The 29-year-old defender takes the armband from Neal Bishop, who left Meadow Lane earlier this summer at the end of his contract.
"I made him captain at the end of last season and he did very well," County manager Chris Kiwomya told the club's official website.
"He's a good guy, a good leader of men and he leads by example.
"He's genuine on the pitch, gives it his best and he was the one that stood out for me."
Leacock, who joined the Magpies from Leyton Orient last summer, added: "At a club like this with the history and everything like that it's an honour.
"I'll be the same person on and off the pitch, just a calm kind of dude.
"I'm happy about the situation and I'm looking forward to it."