Dispersive media refer to a class of natural substances, including living organisms, composite materials, plasma and water, with diverse applications in areas such as biomedicine, microwave sensing, electromagnetic protection, and stealth technology. In the pursuit of investigating the electromagnetic properties of these media, time-domain numerical methods, including finite difference in time domain (FDTD), finite element method, and time domain boundary integral equation method, have been widely utilized. Time-domain numerical methods are preferred to their frequency-domain counterparts owing to their ability to handle nonlinear and wideband problems, as well as various material properties. The FDTD method, in particular, is a highly adaptable, robust, and easy-to-use numerical method that directly solves the Maxwell equations while also simulating the reflection, transmission, and scattering of electromagnetic waves in complex dispersion media. Nonetheless, the traditional FDTD method suffers low computational efficiency arising from the Courant-Friedrichs-Lewy (CFL) stability condition. To solve the problem of low computational efficiency, a new method, the complying divergence implicit finite-difference time-domain (CDI-FDTD) method with a one-step leapfrog scheme, is introduced for lossy Debye dispersive media. The Maxwell equations in the frequency domain form a starting point, and the Fourier transform is utilized to transform the electromagnetic field components from the frequency domain to the time domain. To approximate the integral terms arising from the frequency-to-time domain transformation, a recursive integration (RI) method is employed. Subsequently, the time-domain Maxwell equations and auxiliary variables are discretized with a one-step leapfrog implicit scheme. The iterative formula of the RI-CDI-FDTD algorithm for lossy Debye dispersive media is then derived. The RI-CDI-FDTD method does not change the formulas of the traditional CDI-FDTD method while only requiring to add auxiliary variables for updating field components to the dispersive medium region. The numerical implementation is straightforward, and the electromagnetic modeling is flexible. Moreover, the unconditional stability of the RI-CDI-FDTD algorithm is proven by using the von Neumann method. Finally, some numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed method. In conclusion, our work contributes a crucial numerical simulation tool to accurately modeling complex dispersive media while providing a systemic stability analysis method for time-domain numerical methods.