This paper introduces a novel method for designing approximate circuits by fabricating and exploiting false timing paths, i.e. critical paths that cannot be logically activated. This allows to strongly relax timing constraints while guaranteeing minimal and controlled behavioral change. This technique is applied to an approximate adder architecture, called the Carry Cut-Back Adder (CCBA), in which high-significance stages can cut the carry propagation chain at lower-significance positions. This lightweight approach prevents the logic activation of the carry chain, improving performance and energy efficiency while guaranteeing low worst-case errors. A design methodology is presented along with implementation, error optimization and design-space minimization. The CCBA is proven capable of extremely high accuracy while displaying significant circuit savings. For a worst-case precision of 99.999%, energy savings up to 36% are demonstrated compared to exact adders. Finally, an industry-oriented comparison of 32-bit approximate and truncated adders is carried out for mean and worst-case relative errors. The CCBA outperforms both state-of-the-art and truncated adders for high-accuracy and low-power circuits, confirming the interest of the proposed concept to help building highly-efficient approximate or precision-scalable hardware accelerators.