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In this paper, we consider an infinite-dimensional extension of Chua's circuit (Fig. 1) obtained by replacing the left portion of the circuit composed of the capacitance C2 and the inductance L by a lossless transmission line as shown in Fig. 2. As we shall see, if the remaining capacitance C1 is equal to zero, the dynamics of this so-called time-delayed Chua's circuit can be reduced to that of a scalar nonlinear difference equation. After deriving the corresponding 1-D map, it will be possible to determine without any approximation the analytical equation of the stability boundaries of cycles of every period n. Since the stability region is nonempty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the "period-adding" phenomenon where every two consecutive cycles are separated by a chaotic region.