Alexander stopped short of ruling out a British vote should the French vote no May 29 but acknowledged it would have "serious consequences."
He denied there was any confusion over the government's position despite Tory accusations being in a "shambles" over the issue.
Earlier, Foreign Secretary Jack Straw had suggested Britain would be legally bound to hold a referendum unless the European Council lifted that obligation.
Conservative Shadow Foreign Secretary Liam Fox said the British people should be granted a say regardless of the French result and called on the government to name a date.
But Liberal Democrat Foreign Affairs spokesman Sir Walter Menzies-Campbell said it would be "political madness" to have a referendum if there was no constitution to vote on. He said Europe would have to go "back to the drawing board" if France voted no.
It is likely that the Council of Ministers would decide at their June meeting whether the treaty was doomed.