The stable capillary growth conditions for pulling, from melt, 500-mm-long fiber crystals with constant diameter in the region 150-270 μm by the micro-pulling-down method are investigated. As a model, a substance with the sapphire/YAG eutectic composition is used. The experimental results are compared with theoretical predictions. In order to obtain long fibers with a fine eutectic microstructure, these studies were focused on the meniscus stability at high growth rates of up to 20 mm min-1. A well-adapted relation between heating power and pulling velocity was ascertained. The results are also generally applicable for fiber crystal growth of other materials.