A strip-winding apparatus, comprising: a supply device which supplies a strip to a wound body including an inclined part (23, 24) whose diameter gradually decreases toward outside in an axial direction of the wound body, the wound body being rotatable on an axial line; a rotatable winding roller in which a rotational axis line thereof is capable of inclining with respect to one plane including the axial line of the wound body, the winding roller being capable of pressing the strip supplied to the wound body on the wound body; a movement device (44, 56) which moves the winding roller along an outer circumferential surface of the wound body within the one plane so that the strip is wound on the outer circumferential surface of the wound body in a helical pattern; an inclination angle-changing device which adds a rotational driving force to the winding roller and is capable of changing a roller inclination angle of the rotational axis line of the winding roller with respect to the one plane; and a controller which controls the inclination angle-changing device and is configured to set the roller inclination angle at the time the strip is wound around the inclined part (23, 24) to be greater than the roller inclination angle at the time the strip is wound around a maximum diameter position of the wound body; characterised in that the inclination angle-changing device includes: a speed reducer disposed in the middle of a supply path for the strip and provided with a through-hole through which the strip in the middle of supply passes; and a driving mechanism which adds a rotational driving force to the speed reducer; and wherein the winding roller is supported by an output side of the speed reducer. a supply device which supplies a strip to a wound body including an inclined part (23, 24) whose diameter gradually decreases toward outside in an axial direction of the wound body, the wound body being rotatable on an axial line; a rotatable winding roller in which a rotational axis line thereof is capable of inclining with respect to one plane including the axial line of the wound body, the winding roller being capable of pressing the strip supplied to the wound body on the wound body; a movement device (44, 56) which moves the winding roller along an outer circumferential surface of the wound body within the one plane so that the strip is wound on the outer circumferential surface of the wound body in a helical pattern; an inclination angle-changing device which adds a rotational driving force to the winding roller and is capable of changing a roller inclination angle of the rotational axis line of the winding roller with respect to the one plane; and a controller which controls the inclination angle-changing device and is configured to set the roller inclination angle at the time the strip is wound around the inclined part (23, 24) to be greater than the roller inclination angle at the time the strip is wound around a maximum diameter position of the wound body; characterised in that a speed reducer disposed in the middle of a supply path for the strip and provided with a through-hole through which the strip in the middle of supply passes; and a driving mechanism which adds a rotational driving force to the speed reducer; and wherein the winding roller is supported by an output side of the speed reducer.