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 $26.1 pm 8.7$ $mu {rm G}$. The critical mass of B68, evaluated using both magnetic and thermal/turbulent support, is $M_{rm cr} = 2.30 pm 0.20$ ${rm M}_{odot}$, which is consistent with the observed core mass of $M_{rm core}=2.1$ 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.
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