Catalyzes the attachment of L-aspartate to tRNA(Asp) in a two-step reaction: L-aspartate is first activated by ATP to form Asp-AMP and then transferred to the acceptor end of tRNA(Asp). Is specific for tRNA(Asp) since it aspartylates tRNA(Asn) 3 orders of magnitude less efficiently than tRNA(Asp). ATP + L-aspartate + tRNA(Asp) = AMP + diphosphate + L-aspartyl-tRNA(Asp) kcat is 0.77 sec(-1) for tRNA(Asp) aspartylation at 37 degrees Celsius and 2.7 sec(-1) at 70 degrees Celsius. kcat is 0.12 sec(-1) for tRNA(Asn) aspartylation at 70 degrees Celsius. Homodimer. Constitutively expressed in a constant ratio along the growth of the bacterium. Belongs to the class-II aminoacyl-tRNA synthetase family. Type 1 subfamily.