This note answers a question asked in the electronic abstract of V. Komornik and P. Loreti [Am. Math. Mon. 105, 636-639 (1998; Zbl 0918.11006)]. It proves the transcendence of the number \(q\), which is the smallest \(q\in (1,2)\) for which there exists a unique expansion of 1 as \(1= \sum\delta_n q^{-n}\), with \(\delta_n\in \{0,1\}\) (the sequence \((\delta_n)\) is also known as the Prouhet-Thue-Morse sequence). The proof is a direct consequence of K. Mahler [Math. Ann. 101, 342-366 (1929; JFM 55.0115.01); Corrigendum 103, 532 (1930; JFM 56.0185.02)]. See also for connected results [J.-P. Allouche and M. Cosnard, C. R. Acad. Sci., Paris, Sér. I 296, 159-162 (1983; Zbl 0547.58027) and Acta Math. Hung. 91, 325-332 (2001)].