We use ultrafast electron crystallography to study structural changes induced in graphite by a femtosecond laser pulse. At moderate fluences of < or =21 mJ/cm2, lattice vibrations are observed to thermalize on a time scale of approximately 8 ps. At higher fluences approaching the damage threshold, lattice vibration amplitudes saturate. Following a marked initial contraction, graphite is driven nonthermally into a transient state with sp3-like character, forming interlayer bonds. Using ab initio density functional calculations, we trace the governing mechanism back to electronic structure changes following the photoexcitation.