A convertible is provided with a roof construction which has at least one roof part and a rear window part. The construction can be at least partially deposited in a trunk space in an opening position in the rear area of the vehicle under a rear opening hood. To provide an opening which exposes the trunk space toward the rear, the rear opening hood is laterally linked, in each case by a guiding linkage, along a first forward swivelling axis to a lateral auxiliary frame. Two lateral auxiliary frames are supported on the vehicle body side in the lower rear area of the vehicle behind the first swivelling axis, in each case, by an additional control arm which, by way of a point of rotation, is connected with the pertaining auxiliary frame. At least one of the two auxiliary frames can be connected by way of a releasable locking element with the motor vehicle body. In the rear area, the rear opening hood is connected by a second rearward swivelling axis with the vehicle body for depositing the roof construction. The second swivelling axis is provided with a locking device for the rear opening hood. This locking device can be released for providing an opening which exposes the trunk space toward the rear.