Epitaxial films of hcp Tb(0001) on bcc W(110) substrates are prepared and their geometric, electronic and magnetic properties are studied. The atomic flatness and cleanness and the single-crystalline state of the W(110) substrates and the Tb films are checked using scanning tunneling microscopy (STM), Auger electron spectroscopy (AES) and low-energy and high-energy electron diffraction (LEED and RHEED). Using electron capture spectroscopy (ECS), the surface electron spin polarization of the Tb(0001) films is studied as a function of temperature. The topmost layer is found to order ferromagnetically below a surface Curie temperature T\sbC\sbs = 249.96 K, which lies above both the bulk Curie and Neel temperatures T\sbC\sbb = 220 K and T\sbN\sbb = 228 K, respectively, and exhibits strongly non-monotonic behavior close to 245 K. Novel critical behavior is observed near T\sbC\sbs, suggesting strong surface anisotropies, in accord with a recently predicted and previously unobserved surface phase transition.