In this work, a transportable ultra-stable laser system based on a Fabry-Pérot cavity with crystalline aluminium gallium arsenide (Al₀․₉₂Ga₀․₀₈As) / gallium arsenide (GaAs) mirror coatings, fused silica glass mirror substrates and a 20 cm-long ultra low expansion glass spacer was designed and built to serve as a clock laser for a ⁸⁷Strontium (Sr) lattice clock. The laser system uses an external-cavity diode laser, which is stabilized to a resonance frequency of the Fabry-Pérot cavity using the Pound-Drever-Hall method. This reduces the laser's fractional frequency instability down to the cavity's fractional length instability. Due to the high absorbance of Al₀․₉₂Ga₀․₀₈As/GaAs mirror coatings for visible light, the laser is operated at a wavelength of 1397 nm, which is twice the transition wavelength of a ⁸⁷Sr lattice clock. The laser system therefore includes frequency doubling and light distribution for operation of a ⁸⁷Sr lattice clock. The fundamental limit of the cavity's fractional length instability and thus the laser's fractional frequency instability is determined by the thermal noise floor resulting from Brownian, thermoelastic and thermorefractive noise of the cavity components. The calculated thermal noise floor limit given as modified Allan deviation of the fractional frequency instability mod σᵧ is below 1 · 10⁻¹⁶. Besides the thermal noise, technical noise caused by seismic noise, residual amplitude modulation, laser power, pressure, optical path length and temperature fluctuations affects the laser's fractional frequency instability. The single contributions of the technical noise were investigated and their impact on the laser's fractional frequency instability were suppressed below the thermal noise floor for averaging times around one second using passive or active stabilization. The laser system achieves an instability as low as mod σᵧ = 1.6 · 10⁻¹⁶, which is already a factor 1.3 lower than the theoretically possible instability of mod σᵧ = 2 · 10⁻¹⁶ for the same resonator with tantalum pentoxide (Ta2O5) / fused silica (SiO2) mirrors. This is the lowest fractional frequency instability among published transportable laser systems. Depending on the averaging time of interest, the fractional frequency instability has been reduced by a factor of up to seven compared to Physikalisch-Technische Bundesanstalt (PTB)'s current transportable laser system, which had the lowest fractional frequency instability until now. This reduced instability allows a reduction of the Dick effect limit by roughly a factor of four for interrogation times below 0.5 s, which would reduce the clock's instability limit significantly.