When the consul for 45 BC died on his last day in office, Caesar appointed Gaius Caninius Rebilus, who had been legate at the siege of Alesia, to serve the remainder of his term. Cicero, in his Letters to Friends (VII.30), provides the details.
“At one o'clock, Caesar announced the election of a consul to serve until 1 January—which was the next morning. So I can inform you that in Caninus' consulship, no one had lunch. Still, nothing untoward occurred while he was consul: such was his vigilance that throughout his consulship, he did not sleep a wink.”
It was an unseemly business and is remarked upon by both Tacitus and Suetonius. When another consul died just before the Kalends of January, in fact, Nero made a point of not replacing him. Cassius Dio (XLIII.46) also disapproved of the affair, where one was “appointed consul, served, and ceased to serve all at the same time.”