In this paper, we demonstrated the possibility of efficient dispersion management using a newly developed asymptotically single-mode hybrid fiber that has anomalous dispersion in the 1 μm spectral range. Of the few internal modes of the hybrid fiber, only one mode can propagate at a length longer than 6 m because of the mode's differential optical absorption. This feature was achieved by inserting a highly absorbing ring layer at the position of the electric field minimum of the hybrid mode. About 10 m of an asymptotically single-mode hybrid fiber with an anomalous dispersion of ∼65 ps/(nm⋅km) at 1064 nm was used for the compression of 7 ps chirped pulses down to 440 fs duration. The use of the same hybrid fiber with a length of 6.5 m for intracavity dispersion management allowed us to demonstrate the stable operation of a ring soliton laser scheme based on a semiconductor satiable absorbing mirror. Pulses with duration of approximately 700 fs, a peak power of 850 W and energy as high as 0.55 nJ were extracted directly from the laser.