(a) Letters of administration, with the will annexed, of the estate of a domiciliary decedent for whom a will has been admitted to probate in accordance with this title, and letters of administration of the estate of a domiciliary decedent for whom no will shall have been admitted to probate in accordance with this title, shall be granted by the Register of Wills of the county in which the decedent was domiciled.
(b) Letters of ancillary administration, with the will annexed, of a nondomiciliary decedent for whom a will has been admitted to probate in accordance with this title, and letters of ancillary administration of the estate of a nondomiciliary decedent for whom no will has been admitted to probate in accordance with this title, shall be granted by the Register of Wills for any county. The administration which shall first be lawfully granted in either case shall extend to all the estate of the decedent within Delaware, and shall exclude the jurisdiction of the Register for any other county.