A method for manufacturing a tire comprising a tire-building step, during which a core is set down in a tire-building station and one or more tire components are placed onto said core in order to form an uncured annular tire during which the core bearing its uncured annular tire is transported from the tire-building station as far as a curing station distant from the tire-building station, a curing step during which the assembly formed by said core and the uncured annular tire placed in the curing station is cured to obtain a cured annular tire, a second transfer step during which the core bearing the cured annular tire is transported from the curing station to an extraction station distant from the curing station, and then an extraction step during which, in the extraction station, the cured annular tire is separated from the core, wherein the core has a tubular wall which extends along and around a central axis and which extends in thickness radially with respect to said central axis from a radially outer first annular face, referred to as “receiving face”, which is intended to receive the tire component or components, as far as a radially inner second annular face referred to as “exchange face” which surrounds the central axis and which is intended to allow exchanges of heat between the tubular wall of the core and thermal sources external to said core, wherein during the curing step a first, heating, thermal source is pressed against the exchange face of the core so as to perform the curing operation, and then said first thermal source is dissociated from the core so as to leave the exchange face free again when the second transfer step of transferring to the extraction station is performed, and wherein, during the extraction step, a second, refrigerating, thermal source is brought into contact with the exchange face thus freed, so as to cool said core in such a way as to create a radial clearance between the receiving face of the core and the cured annular tire through differential radial contraction of the tubular wall of the core relative to the cured annular tire, and the cured annular tire is removed axially from the core.