Electronic textiles demand a new family of flexible circuit boards in the construction of fiber or fiber assemblies. This paper presents a stretchable woven fabric circuit board (FCB) with permanent as well as detachable electrical connections to sensors or other wearable electronics components. The woven FCB was created by integrating conductive yarns into an elastic woven fabric. Permanent connection was designed between the conductive tracks and flexible sensors; detachable connection was achieved by the helical structure of conductive yarns wrapping around the rigid component electrode encapsulated within elastomeric layer. The developed FCB, with its connections to flexible sensors or rigid components, is porous, flexible, and capable of stretching to 30% strain. The woven FCB with permanent connection to temperature sensors has a large fatigue life of more than 10,000 cycles while maintaining constant electrical resistance due to crimped configurations of the conductive track in the elastic fabric substrate and stable contact resistance. A prototype of the FCB assembly, with independent light-emitting diodes electrically linked and mechanically supported by the woven FCB, is also demonstrated for wearable applications.