Context. Exploring the structure and dynamics of cold starless clouds is necessary to understand the different steps leading to the formation of protostars. Because clouds evolve slowly, many of them must be studied in detail to identify different moments in a cloud's lifetime. Aims. We study a fragment of the long filament L1506 in the Taurus region, which we name L1506C, a core with interesting dust properties observed by the PRONAOS balloon-borne telescope. Methods. To trace the mass content of L1506C and its kinematics, we mapped the dust emission, and the line emission of two key species, C 18 O and N 2 H + . We also observed 13 CO and C 17 O. We model the species emission using 1D Monte Carlo models.Results. This cloud is reminiscent of L1498 but also exhibits peculiar features: i) a large envelope emitting solely in 13 CO that contains a much smaller core with strong C 18 O depletion at its centre despite a low maximum opacity (A V ∼ 20 mag); ii) extremely narrow C 18 O lines indicating a low, non-measurable turbulence; iii) contraction traced by C 18 O itself (plus rotation); iv) unexpectedly, the kinematical signature of the external envelope is opposite to that of the core, the 13 CO and C 18 O velocity gradients having opposite directions and the C 18 O line profile being blue peaked in contrast to the 13 CO one which is red peaked. The core is large (r = 3 × 10 4 AU) and not very dense (n(H 2 ) ≤ 5 × 10 4 cm −3 , possibly less). This core is therefore not yet prestellar. Conclusions. All these properties imply that the core is kinematically detached from its envelope and in the process of forming a prestellar core. This is the first time that the dynamical formation of a prestellar core has been witnessed. The extremely low turbulence could explain the strong depletion of this core despite its relatively low density and opacity in contrast to undepleted cores such as L1521E, which exhibits a turbulence at least 4 times as high.