Officials had argued the United States team, which included drug cheat Crystal Cox, should be stripped of their gold with Russia promoted to first place, Jamaica to second and Team GB - with Christine Ohuruogu, Nicola Sanders, Donna Fraser and McConnell - into third.
Following a previous decision to let the result stand, the case was sent back on an appeal last November, with the BOA arguing there had been precedents set in similar cases. Cox, who ran in the Olympic heats but not the final, was stripped of her gold and handed a four-year ban after admitting taking steroids.
But the IOC said a technicality in the rules at that time imposed a statute of limitations on any appeal. That has finally left the Americans in the clear and ended British hopes of gaining what many will argue is sporting justice.
"We're disappointed but the BOA feel it's been taken as far we could go with it," said UK Athletics chairman Ed Warner. "All the advice suggests a legal appeal would fail. We've now informed the girls but it's not what we wanted."