A process for making an optical fiber includes the steps of inserting a rod of the core glass composition into a closed tube made of the cladding glass. The diameter of the rod is substantially less than the inner diameter of the tube. The glass transition temperature of the core glass must be substantially lower than the glass transition temperature of the cladding and the rod is placed in contact with the tube along its entire length. When heat is applied to the lower portion of the rod and tube, the rod melts and forms a thin film on the inner surface of the tube which can rapidly be fined to a relatively pure glass. This melted glass forms a melt pool in the bottom of the tube, and the tube and pool can then be drawn into a fiber with the desired characteristics. A process for making a graded optical fiber utilizing a modified chemical vapor deposition process is also disclosed wherein a cladding glass is entrained on the inner surface of a tube substrate. A core rod is then inserted into the coated substrate tube. At the high drawing temperatures, ion exchange occurs between the core glass and the cladding glass to produce low loss graded optical fibers having an improved refractive index profile. Low loss, single mode optical fibers and single polarization guidance fibers can also be made by this method.