A new Weber-type integral transform and its inverse are defined for the representation of a function that satisfies the Dirichlet and Robin-type boundary conditions, respectively. The orthogonality relations of the transform kernel are derived by using the properties of Bessel functions. The new Weber integral transform of some particular functions is determined. The integral transform defined in the present paper is a suitable tool for determining analytical solutions of transport problems with sliding phenomena that often occur in flows through micro channels, pipes or blood vessels. The heat conduction in an annular domain with Robin-type boundary conditions is studied. The subroutine of the Mathcad software is used to determine the positive roots of the transcendental equation involved in the definition of the new integral transform.