The work is expected to start this month and the project is expecetd to be completed in nine weeks.
The development comes after magistrates formally agreed to 'stop up' the highway rights to the footbridge at a hearing earlier this month.
Councillor Mark Lowry, Cabinet member for Finance and City Centre champion said: "This bridge completely dominates Frankfort Gate. Removing it will open up Frankfort Gate, create extra light and space in this part of the city centre as well as getting rid of a tired and ugly structure."
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Western Approach Car Park and its connecting covered footbridge were built in the late 1980s, with the bridge, ramps and lifts linking the car park to the city centre over Western Approach.
The demolition is part of Plymouth Council's long term plan to regenerate Colin Campbell Court and to link the city centre to the sea.
The short bridge from the car park will be removed first; both the car park and the city centre end of King Street will be closed temporarily. Once the main span is removed, the lift shafts will be dismantled before the paving reinstated.
A new crossing is also being planned for the north bound lane and will be laid diagonally following more closely the walking routes of locals going from Frankfort Gate to King Street.
