The Texas-based company will foot the bill if it fails to find the wreckage.
The disappearance of MH370 remains shrouded in mystery. The Malaysia Airlines flight fell off radar on 8 March 2014, between Kuala Lumpur and Beijing, with 239 people on board.
A massive maritime search operation for the plane cleared 120,000 sq km at an estimated cost of about A$200m (£120m; €133m), before it was suspended in January.
Ocean Infinity has not revealed the estimated cost of the new search. According to Mr Chester, the company will focus on a 25,000 sq km area identified by the Australian Transport Safety Bureau as having a "high probability" of containing the aircraft.
The company is using a centuries-old model known in the salvage industry as "no cure-no pay" - a type of deal usually applied in the recovery of valuable sunken cargo.
Under such a deal, a salvage company will take on the financial risk of a recovery and recoup from the owner a percentage of the cargo's value if it is found, often 80 or 90%.
In this case, Ocean Infinity would likely be working instead for a set fee from the Malaysian government, and for the significant publicity on offer should it find the wreckage, an industry expert told the BBC.
MH370 was carrying passengers and crew from 14 different countries when it disappeared. Most were from China and Malaysia.
Australia led the initial search, after aviation officials identified the ocean floor off its coast as the likely location of the wreckage. The country has agreed to provide technical assistance for the new search, Mr Chester said.
Earlier this month, Malaysian Transport Minister Liow Tiong Lai said the government had received proposals from three private search firms - Ocean Infinity, Dutch firm Fugro and an unidentified Malaysian company.
Delivering its report into the disappearance earlier this month, Australia's Transport Safety Bureau said it was "almost inconceivable" that the aircraft had not been found.
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