The one-particle three-dimensional Dirac equation with spherical symmetry is solved for the Hulthén potential. The s-wave relativistic energy spectrum and two-component spinor wavefunctions are obtained analytically. Conforming to the standard feature of the relativistic problem, the solution space splits into two distinct subspaces depending on the sign of a fundamental parameter in the problem. Unique and interesting properties of the energy spectrum are pointed out and illustrated graphically for several values of the physical parameters. The square integrable two-component wavefunctions are written in terms of the Jacobi polynomials. The nonrelativistic limit reproduces the well-known nonrelativistic energy spectrum and results in Schrödinger equation with a 'generalized' three-parameter Hulthén potential, which is the sum of the original Hulthén potential and its square.