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The purpose of the frequency-stable laser described in this contribution is the use for an optical frequency standard on the basis of a single trapped indium ion, stored in a radio frequency trap. As laser source for excitation of the In+ clock transition , the fourth harmonic of a commercial diode-pumped quasi-monolithic Nd:YAG non-planar ring laser (miser) operating at 946 nm is used. The miser is locked to a Zerodur reference cavity (finesse 70000), which is placed inside a thermo-stabilized vacuum chamber. The whole setup is mounted on an optical breadboard placed on top of an AVI (active vibration isolation) support. In order to characterize the absolute frequency stability of the miser system we use a further independent ULE reference cavity with a finesse of 60 000, which is mounted on a separate and independent AVI platform. A superior way to characterize the frequency stability of the miser is the use of a second independent laser, stabilized to another independent ULE cavity with a finesse of 200000. This second laser system is assembled in another laboratory with a separate basement. The laser light is guided by a 100 m long optical fibre between these two laser systems. The power circulating inside the cavities is stabilized by detecting the transmitted light and applying the corresponding error signal to the AOMs in front of the cavities. The frequency stability of the miser is analyzed by superposing parts of the two independently stabilized lasers and evaluating the corresponding beat signal in both the frequency and time domain. A frequency stability on the Hertz level is then observed. The measured root Allan-variance reaches values around Δν/ν = 10-15 (τ ≈ 1 s). A detailed analysis of the performance of the AVI system as well as experimental studies of the different factors, contributing to the laser frequency instability, is presented.