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 cannot aspartylate tRNA(Asn). ATP + L-aspartate + tRNA(Asp) = AMP + diphosphate + L-aspartyl-tRNA(Asp) Homodimer. Belongs to the class-II aminoacyl-tRNA synthetase family. Type 1 subfamily.