  Cryogenic ultra-stable lasers have extremely low thermal noise limits and
frequency drifts, but they are more seriously affected by vibration noise from
cryostats. Main material candidates for cryogenic ultra-stable cavities include
silicon and sapphire. Although sapphire has many excellent properties at low
temperature, the development of sapphire-based cavities is less advanced than
that of silicon-based. Using a homemade cryogenic sapphire cavity, we develop
an ultra-stable laser source with a frequency instability of
$1.9\times10^{-16}$. This is the best frequency instability level among similar
systems using cryogenic sapphire cavities reported so far. Low vibration
performance of the cryostat is demonstrated with a two-stage vibration
isolation, and the vibration suppression is further improved by different
mixing ratio of the gas-liquid helium. With this technique, vibrations at
frequencies higher than tens of hertz are greatly suppressed.
