N.J. Stat. § 40:55D-34
For purpose of preserving the integrity of the official map of a municipality no permit shall be issued for any building or structure in the bed of any street or public drainage way, flood control basin or public area reserved pursuant to section 23 of P.L. 1975, c.291 (C.40:55D-32) as shown on the official map, or shown on a plat filed pursuant to this act before adoption of the official map, except as herein provided. Whenever one or more parcels of land, upon which is located the bed of such a mapped street or public drainage way, flood control basin or public area reserved pursuant to section 23 of P.L. 1975, c.291 (C.40:55D-32), cannot yield a reasonable return to the owner unless a building permit is granted, the board of adjustment, in any municipality which has established such a board, may, in a specific case, by an affirmative vote of a majority of the full authorized membership of the board, direct the issuance of a permit for a building or structure in the bed of such mapped street or public drainage way or flood control basin or public area reserved pursuant to section 23 of P.L. 1975, c.291 (C.40:55D-32), which will as little as practicable increase the cost of opening such street, or tend to cause a minimum change of the official map and the board shall impose reasonable requirements as a condition of granting the permit so as to promote the health, morals, safety and general welfare of the public. Sections 59 through 62 of P.L. 1975, c.291 (C.40:55D-72 through C.40:55D-75) shall apply to applications or appeals pursuant to this section. In any municipality in which there is no board of adjustment, the planning board shall have the same powers and be subject to the same restrictions as provided in this section.
The board of adjustment shall not exercise the power otherwise granted by this section if the proposed development requires approval by the planning board of a subdivision, site plan or conditional use in conjunction with which the planning board has power to direct the issuance of a permit pursuant to subsection b. of section 47 of P.L. 1975, c.291 (C.40:55D-60).
N.J.S. § 40:55D-34