The argument is that to allow such notifications to be made at an embassy would breach the "inviolability" of embassies and consulates that is guaranteed under the Vienna Convention on Diplomatic Relations.

The Trump administration argues as well that if Sudan and other countries can be served notice of lawsuits at embassies, the U.S. could suffer a similar fate at its embassies.

But lawyers for the Cole victims counter that the U.S. has a firm policy of never accepting such legal service at its embassies and that the U.S. has not had any difficulty enforcing that policy over a period of more than four decades.

On Wednesday when the case was argued, a grim group of Cole survivors and relatives of the dead sat in the chamber.