We studied the shape transformation (by use of TEM and optical absorption spectroscopy) of gold nanorods in micellar solution by exposure to laser pulses of different pulse width (100 fs and 7 ns) and different energies (mu J to mJ) at 800 nm, where the longitudinal surface plasmon oscillation of the nanorods absorb. At moderate energies, the femtosecond irradiation melts the nanorods to near spherical particles of comparable volumes while the nanosecond pulses fragment them to smaller near-spherical particles. At high energies, fragmentation is also observed for the femtosecond irradiation. A mechanism involving the rate of energy deposition as compared to the rate of electron-phonon and phonon-phonon relaxation processes is proposed to determine the final fate of the laser-exposed nanorods, i.e., melting or fragmentation.