The α-decay rates for the nuclides 168,170,171,172,173,174,176Yb and 148,150,152,154Gd have been estimated from transmission probabilities in a systematic α-nucleus potential and from an improved fit to α-decay rates in the rare-earth mass region. Whereas α-decay of 152Gd in natural gadolinium is a severe obstacle for the use of gadolinium as a low-energy solar-neutrino detector, we show that α-decay does not contribute significantly to the background in a ytterbium detector. An extremely long α-decay lifetime of 168Yb is obtained from calculation, which may be close to the sensitivity limit in a low-background solar-neutrino detector.