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) Binds 3 Mg(2+) cations per subunit. The strongest magnesium site (Mg1) is bound to the beta- and gamma-phosphates of ATP and four water molecules complete its coordination sphere. kcat is 0.061 sec(-1) for tRNA(Asp) aspartylation (at 60 degrees Celsius). Optimum temperature is 60 degrees Celsius. Homodimer. Belongs to the class-II aminoacyl-tRNA synthetase family. Type 2 subfamily.