Measurements of the specific heat of quasi one-dimensional conductors, (TaSe 4 ) 2 I and (NbSe 4 ) 3 I, between 0.1 and 8 K, are reported. Whereas no deviation from the P law occurs at low temperature for (NbSe 4 ) 3 I, we have measured two anomalies in (TaSe^I: an excess heat capacity with a peak at J= 1.75 K and a specific heat which follows a P 2 law below 0.4 K. We ascribe this excess heat capacity to the charge-density-wave phase excitations above their pinning frequency in the insulating Peierls state (rp = 263 K).PACS numbers: 65.40.Em, 63.20.Dj, 63.50, + x, 72.15.Nj It is now well known that many few-dimensional compounds are unstable and undergo a Peierls transition at a critical temperature, T ? } This instability is characterized by the opening of an electronic gap at the Fermi level, by a softening in the phonon spectrum at 0==2/c F with A: F the Fermi wave vector, and by a modulation of the electron charge density with the same periodicity called a charge-density wave (CDW). Below T ? the soft-phonon branch splits into two different modes 2 " 4 : an opticlike amplitude mode and an acousticlike phase mode. For ideal one-dimensional conductors, when Q is incommensurate with the lattice, the phase mode or phason exists at zero frequency and corresponds to the current provided by the sliding motion of the CDW as proposed by Frohlich. 5 In real systems, as shown by Lee, Rice, and Anderson, 3 the gapless character of the phase mode is destroyed through various mechanisms, especially impurity pinning.Recently, the interest has focused on several families of one-dimensional inorganic compounds undergoing Peierls transitions, namely, transition-metal trichalcogenides (NbSe 3 , TaS 3 ), blue bronze, and halogenated transition-metal tetrachalcogenide (TaSe^L 6 In these compounds an extra dc current appears when the applied electric field is higher than a threshold value (typically 0.1-1 V/cm). This extra conductivity has been ascribed to Frohlich-type conduction when the CDW becomes unpinned and moves through the crystal. 6 By complex-impednace measurements in the millimeter-wave range the pinned sliding model for NbSe 3 and TaS 3 is found to be overdamped with a spectral response in the megahertz-gigahertz range, 7 whereas for (TaSe^I this mode is underdamped, indicating a pinning frequency 8 of 35 GHz or 1.7 K. The frequency of this pinned mode is just in the right frequency range to allow for the detection of CDW excitations by thermodynamical measurements.Hereafter we report on heat-capacity measurements on (TaSe^I and, for comparison, on (NbSe 4 ) 3 I. These compounds consist of parallel MSe 4 chains well separated from one another by halogen atoms. In each MSe 4 chain the metal atom is located at the center of a rectangular antiprism of eight Se atoms. The d z2 band filling of the transition metal M can be varied according to the composition. This change in band filling leads to quite different structural and electrical properties: (TaSe 4 ) 2 I and (NbSe 4 )ioI 3 undergo a Peierls transition at T P ...