  Filamentary structures are common in molecular clouds. Explaining how they
fragment to dense cores is a missing step in understanding their role in star
formation. We perform a case study of whether low-mass filaments are close-to
hydrostatic prior to their fragmentation, and whether their fragmentation
agrees with gravitational fragmentation models. For this, we study the 6.5 pc
long Musca molecular cloud that is an ideal candidate for a filament at an
early stage of fragmentation. We employ dust extinction mapping in conjunction
with near-infrared data from the NEWFIRM instrument, and 870 um dust continuum
emission data from the LABOCA instrument, to estimate column densities. We use
the data to identify fragments from the cloud and to determine the radial
density distribution of its filamentary part. We compare the cloud's morphology
with 13CO and C18O line emission observed with the APEX/SHeFI instrument. The
Musca cloud is pronouncedly fragmented at its ends, but harbours a remarkably
well-defined, 1.6 pc long filament in its Center region. The line mass of the
filament is 21-31 Ms pc^-1 and FWHM 0.07 pc. Its radial profile can be fitted
with a Plummer profile that has the power-index of 2.6 \pm 11%, flatter than
that of an infinite hydrostatic filament. The profile can also be fitted with a
hydrostatic cylinder truncated by external pressure. These models imply a
central density of 5-10 x 10^4 cm^-3. The fragments in the cloud have a mean
separation of 0.4 pc, in agreement with gravitational fragmentation. These
properties, together with the subsonic and velocity-coherent nature of the
cloud, suggest a scenario in which an initially hydrostatic cloud is currently
gravitationally fragmenting. The fragmentation has started a few tenths of a
Myr ago from the cloud ends, leaving its center yet relatively non-fragmented,
possibly because of gravitational focusing in a finite geometry.
