A layer-by-layer growth of ultrathin films right from the beginning would be desirable for establishing a straightforward correlation between magnetism and structure. We give experimental evidence that with the help of pulsed laser deposition (PLD) we can achieve layer-by-layer growth for Fe on Cu(100) in contrast to deposition by molecular-beam epitaxy. We present the results of a comprehensive study of the structural and magnetic properties of PLD-grown ultrathin Fe films of thicknesses between 2 and 10 monolayers (ML) deposited at room temperature. We show scanning tunneling microscopy images and low-energy electron-diffraction (LEED) patterns as well as intensity vs energy (IV) LEED curves demonstrating that PLD-grown Fe/Cu(100) has an isotropic fcc structure. We characterize the magnetic properties of our films by the magnetooptical Kerr effect. Following the improved growth and morphology, we found strong differences in the magnetic behavior of these films in comparison with Fe thermally deposited onto Cu(100): PLD-grown ultrathin Fe/Cu(100) shows an in-plane easy axis of magnetization in the thickness range 2-5 ML and again from about 10 ML. where the film structure is dominated by the bcc-Fe bulk phase, while there is a perpendicular easy axis of magnetization between 7 and 10 ML coverage. These results are discussed in terms of the different growth and structure due to the characteristic features of the PLD technique.