Atom Trap Trace Analysis (ATTA) of 39Ar at the Kirchhoff-Institute for Physics in Heidelberg is a laser-based counting method for the rare argon isotope 39Ar. Due to its half-life of 269 a it is ideal for dating water and ice samples within the range from 50 to 1000 years. This diploma thesis summarises the activities at the ATTA-experiment in 2012. Besides an extensive analysis of the atomic beam, tools for permanent characterisation and monitoring of the apparatus were developed. By optimising the transversal and longitudinal cooling as well as the magneto-optical trap (MOT) an increase of the 39Ar-counting rate for an atmospheric sample by a factor of 7 up to 4:1(3) atoms/h was possible. A contribution to the counting rate based on contamination by measurements with enriched samples could be limited to 0:3 atoms/h. Furthermore, the stability of the counting rate was significantly improved. A long-term measurement over 12 h showed a Poissonian behaviour for an enriched sample. The statistics of an atmospheric reference measurement over in total 48 h spread over 2 weeks for calibrating the modern standard could also be demonstrated to follow a Poisson distribution. Finally, ATTA for 39Ar was the first time used successfully by dating two natural water samples to 377(66) and 360(68) years.