Lagrangian methods play a very important role in computational fluid dynamics and especially suitable for dealing with the physical problems related to high-intensity magnetic field such as Z-pinch, Tokamak, ICF, and so on. In these physical problems the density and thermal pressure are always non-negative. However, such positivity property is not always satisfied by approximated solutions which obtained by a numerical scheme. To deal with this problem, the paper develops a Lagrangian HLLD approximate Riemann solver which can keep positivity-preserving property under some appropriate signal speeds. With this solver, a conservative Lagrangian scheme for solving the ideal compressible magnetohydrodynamics equations in one-dimensional is proposed. At last, some numerical examples are presented to demonstrate the positivity-preserving property of our scheme.