The magnetic field structure , kinematical stability , and evolutionary status of the starless dense core Barnard 68 ( B68 ) are revealed based on the near-infrared polarimetric observations of background stars , measuring the dichroically polarized light produced by aligned dust grains in the core . After subtracting unrelated ambient polarization components , the magnetic fields pervading B68 are mapped using 38 stars and axisymmetrically distorted hourglass-like magnetic fields are obtained , although the evidence for the hourglass field is not very strong . On the basis of simple 2D and 3D magnetic field modeling , the magnetic inclination angles on the plane-of-sky and in the line-of-sight direction are determined to be 47 ^ { \circ } \pm 5 ^ { \circ } and 20 ^ { \circ } \pm 10 ^ { \circ } , respectively . The total magnetic field strength of B68 is obtained to be 26.1 \pm 8.7 \mu { G } . The critical mass of B68 , evaluated using both magnetic and thermal/turbulent support , is M _ { cr } = 2.30 \pm 0.20 { M } _ { \odot } , which is consistent with the observed core mass of M _ { core } = 2.1 M _ { \odot } , suggesting nearly critical state . We found a relatively linear relationship between polarization and extinction up to A _ { V } \sim 30 mag toward the stars with deepest obscuration . Further theoretical and observational studies are required to explain the dust alignment in cold and dense regions in the core .