Ultrathin Fe films on Cu(111) have been grown by pulsed-laser deposition (PLD) whose instantaneous deposition rate is about six orders of magnitude larger than that of the conventional molecular-beam epitaxy based on thermal deposition (TD). Compared to the TD prepared Fe/Cu(111) films, the PLD films have a significantly improved layer-by-layer morphology and a substantially enhanced stability of the fcc phase as characterized by scanning tunneling microscopy and electron-diffraction techniques. The magnetic properties of the PLD films, investigated by magneto-optical Kerr effect, also show remarkably different behavior as compared to the TD films. At low thickness (<3 ML), while the TD films are characterized by a low net moment and a short-range ferromagnetic order, the PLD films show a high net moment with a true long-range ferromagnetic order. Above 3 ML both PLD and TD films undergo a magnetic transition though with apparently different structural origin. We discuss these results based on the different growth and structure between the PLD and TD grown Fe films.