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. 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. To trace the mass content of L1506C and its kinematics, we mapped the dust emission, and the line emission of two key species, C
18O and N
2H
+. We also observed
13CO and C
17O. We model the species emission using 1D Monte Carlo models. This cloud is reminiscent of L1498 but also exhibits peculiar features: i) a large envelope emitting solely in
13CO that contains a much smaller core with strong C
18O depletion at its centre despite a low maximum opacity (A
V∼20mag); ii) extremely narrow C
18O lines indicating a low, non-measurable turbulence; iii) contraction traced by C
18O itself (plus rotation); iv) unexpectedly, the kinematical signature of the external envelope is opposite to that of the core, the
13CO and C
18O velocity gradients having opposite directions and the C
18O line profile being blue peaked in contrast to the
13CO one which is red peaked. The core is large (r=3x10
4AU) and not very dense (n(H
2)≤5x10
4/cm
3, possibly less). This core is therefore not yet prestellar. 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.