President Abdul Hamid has "signed the paper appointing the new chief justice," a Bangabhaban presidential palace spokesman told PTI.
A Law Ministry statement said 64-year-old Sinha would take over on January 17, as present Chief Justice Muzammel Hossain will retire on January 16.
Sinha was the senior most judge in the apex Appellate Division of the Supreme Court after Hossain.
Unitary Bangladesh's Supreme Court comprises two divisions - the Appellate Division and the High Court division.
Justice Sinha is known for a number of landmark judgments including those on the killing of Bangabandhu Sheikh Mujibur Rahman and the 5th and 13th amendments to the constitution.
Then he enrolled as an advocate of the High Court and the Appellate Division in 1978 and 1990 respectively.
He was appointed as judge of the High Court in 1999 and as judge of the Appellate Division in 2009.