No injuries were reported.
There has been minor damage to the parapet of the Grade II listed bridge. It is currently being made safe with temporary fencing being installed and the road will be reopened within an hour or two.
A Wiltshire Police spokesman said: “At around 4.20am today (May 18) we received a call regarding a single-vehicle road traffic collision on the High Street in Malmesbury.
“It involved a black Renault and there was some damage to the bridge, which caused the road to be closed.
“The driver, a 41-year-old woman from Bristol, was arrested on suspicion of drug driving and is currently in custody at Melksham Police Station.
“Nobody else was in the vehicle.”
A fire service spokeswoman: "A crew from Malmesbury were called to High Street, Malmesbury at 4.26am this morning following a single car colliding with a bridge.
"There was no fire so our crew assisted in making the scene safe and handed the incident over to the police.
"The stop came at 5.55am."