The instrumentation for VLT/VLTI 1 facility of the European Southern Observatory at Paranal (Chile)includes the infrared beam-combiner called AMBER, that covers the near infrared bands up to 2.5 μm. The cold spectrograph we describe is the AMBER subsystem responsible of wavelength analysis and several other functions, all of them performed by means of optics, analyzers, and mechanisms working at the temperature of liquid nitrogen boiling at atmospheric pressure. The cryo-mechanical design of the spectrograph we describe here used extensively the methods of finite element analysis and the laboratory tests validated this approach. The final optical quality we measured in the laboratory before shipping the instrument to Grenoble or integration (December 2002),is well inside the specification the AMBER staff assigned to the spectrograph. Simulations show that its total contribution to visibility loss of AMBER is less than 2%.