This paper presents a unified equation of state (EOS) for dense matter in neutron stars, covering both the crust and the liquid core. The EOS is derived using the Skyrme-type effective nuclear interaction SLy, which is particularly suitable for describing very neutron-rich matter. The structure of the crust is calculated in the $T=0$ approximation under the assumption of ground state composition, while the EOS of the liquid core is calculated assuming minimal $n\mu\mu$ composition. The crust-core transition is found to be a weak first-order phase transition with a relative density jump of about one percent. Parameters of static neutron stars are calculated and compared with observational data, yielding a minimum mass of 0.094 M$_\odot$ and a maximum mass of 2.05 M$_\odot$. The effects of rotation on the minimum and maximum masses are briefly discussed. The EOS and neutron star models are constructed using the SLy interaction, and their properties are compared with those obtained using older FPS effective nuclear interactions. The paper also includes a detailed description of the method used to solve the many-body problem, the structure and EOS of the crust, and the EOS of the liquid core. The results are presented in tables and figures, and the paper concludes with a summary and discussion of the findings.This paper presents a unified equation of state (EOS) for dense matter in neutron stars, covering both the crust and the liquid core. The EOS is derived using the Skyrme-type effective nuclear interaction SLy, which is particularly suitable for describing very neutron-rich matter. The structure of the crust is calculated in the $T=0$ approximation under the assumption of ground state composition, while the EOS of the liquid core is calculated assuming minimal $n\mu\mu$ composition. The crust-core transition is found to be a weak first-order phase transition with a relative density jump of about one percent. Parameters of static neutron stars are calculated and compared with observational data, yielding a minimum mass of 0.094 M$_\odot$ and a maximum mass of 2.05 M$_\odot$. The effects of rotation on the minimum and maximum masses are briefly discussed. The EOS and neutron star models are constructed using the SLy interaction, and their properties are compared with those obtained using older FPS effective nuclear interactions. The paper also includes a detailed description of the method used to solve the many-body problem, the structure and EOS of the crust, and the EOS of the liquid core. The results are presented in tables and figures, and the paper concludes with a summary and discussion of the findings.