Method for converting a container stream delivered via a first transport section in a first transport direction into a second container stream further conveyed on a second transport section in a second transport direction, wherein the first container stream and the second container stream have different widths perpendicularly to the first and second transport directions (A, C), using a conversion section disposed between the transport sections, wherein the containers being conveyed, with at least one container receiver (5, 5a, 5b), are gripped by a container gripper on the conversion section at least at one first transfer position of the first transport section, and, with this container receiver (5, 5a, 5b), are transported in a third transport direction to different second transfer positions, which are provided in such a way as to be distributed on the second transport section transverse to the second transport direction, and are handed over there to the second transport section, wherein the at least one container receiver (5, 5a, 5b) is formed by a carrier guided on a stator of the conversion section, and by a container gripper at the carrier, wherein the container receivers are at the same time also configured for a translational movement of the container gripper, in that, at the handover of the respective container to a track (7.1 - 7.4) of the second transport section, the container concerned, initially with the container receiver, is moved further in the second transport direction laterally by the carrier, for which purpose each container receiver is provided with an arm, which (a) is provided at the carrier such as to be displaceable in a controlled manner axially and radially to the container axis, and carries the container gripper, or wherein (b) the handover to the transport section takes place by a rotational movement of the respective arm at the associated carrier, in that the respective arm is rotated or pivoted in a controlled manner about a vertical axis, characterised in that the circumferential surface or casing surface of the respective container is supported by a support surface (16.1) of a container support element provided at the container receiver, which forms a preferably cambered support surface (16.1) for the circumferential surface or casing surface of the respective container, and that the container support element can be pivoted in a controlled manner about a vertical or essentially vertical axis. (a) is provided at the carrier such as to be displaceable in a controlled manner axially and radially to the container axis, and carries the container gripper, or wherein (b) the handover to the transport section takes place by a rotational movement of the respective arm at the associated carrier, in that the respective arm is rotated or pivoted in a controlled manner about a vertical axis, characterised in that the circumferential surface or casing surface of the respective container is supported by a support surface (16.1) of a container support element provided at the container receiver, which forms a preferably cambered support surface (16.1) for the circumferential surface or casing surface of the respective container, and that the container support element can be pivoted in a controlled manner about a vertical or essentially vertical axis.