Context. 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. Aims. 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. To accomplish this, we study the ~6.5 pc long Musca molecular cloud, which is an ideal candidate for a filament at an early stage of fragmentation. Methods. We employ dust extinction mapping, in conjunction with near-infrared JHKS-band data from the CTIO/NEWFIRM instrument, and 870 μm dust continuum emission data from the APEX/LABOCA instrument to estimate column densities in Musca. 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. Results. The Musca cloud is pronouncedly fragmented at its ends, but harbors a remarkably well-defined, ~1.6 pc long filament in its center region. The line mass of the filament is 21–31 M⊙ pc-1 and the full width at half maximum (FWHM) 0.07 pc. The radial profile of the filament can be fitted with a Plummer profile, which has the power-index of 2.6 ± 11% and is 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 × 104 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 started a few tenths of a Myr ago from the ends of the cloud, leaving its center still relatively nonfragmented, possibly because of gravitational focusing in a finite geometry.