Exhibit 99.1350CERT

Certification Pursuant to Rule 30a-2(b) under the 1940 Act and

Section 906 of the Sarbanes Oxley Act

Pursuant to 18 U.S.C. § 1350, the undersigned officer of BlackRock Core Bond Trust (the “registrant”), hereby certifies, to the best of her knowledge, that the registrant's Report on Form N-CSR for the period ended February 28, 2010, (the “Report”) fully complies with the requirements of Section 13(a) of the Securities Exchange Act of 1934, as amended, and that the information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the registrant.

Date: April 28, 2010

/s/ Anne F. Ackerley

Anne F. Ackerley

Chief Executive Officer (principal executive officer) of

BlackRock Core Bond Trust

Pursuant to 18 U.S.C. § 1350, the undersigned officer of BlackRock Core Bond Trust (the “registrant”), hereby certifies, to the best of his knowledge, that the registrant's Report on Form N-CSR for the period ended February 28, 2010, (the “Report”) fully complies with the requirements of Section 13(a) of the Securities Exchange Act of 1934, as amended, and that the information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the registrant.

Date: April 28, 2010

/s/ Neal J. Andrews

Neal J. Andrews

Chief Financial Officer (principal financial officer) of

BlackRock Core Bond Trust

This certification is being furnished pursuant to Rule 30a-2(b) under the Investment Company Act of 1940, as amended, and 18 U.S.C. Section 1350 and is not being filed as part of the Form N-CSR with the Securities and Exchange Commission.