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The noble gas radioisotope 39Ar with a half-life of 269 years is an almost ideal tracer for dating ice and water samples in the time range of 50 to 1000 years ago, for which no other reliable methods exist. Due to its very low relative abundance of 39Ar/Ar = 8.1(3)*10^-16, 39Ar has only been routinely measured by Low-Level Counting so far. However, since Low-Level Counting requires samples in the order of 1000 L, the application of 39Ar, besides some proof-of-principle experiments, has been limited to groundwater studies until now. In the course of this experimental thesis, the required sample size could be reduced down to 2mLSTP of argon by further development of the atom-optical analytical method Atom Trap Trace Analysis (ATTA). This amount of gas can be extracted from ~5 kg water or ice. Due to a redesign of the laser system together with an optimisation of the vacuum system, the atmospheric count rate could be moreover doubled up to 7.0 atoms per hour. This enables 39Ar-measurements with a typical relative uncertainty of 10% within one day. The measuring technique was validated in an intercomparison study based on slightly enriched samples with the well established Low-Level Counting Laboratory in Bern as well as with a new laboratory at the Pacific Northwest National Laboratory. Furthermore, the new method was verified by analysing samples with a known subatmospheric 39Ar concentration. A defined sample with 10% of the modern concentration could be distinguished with 1-sigma confidence from the background. In a first application, ice samples were analysed for 39Ar by ATTA for the first time. Three glacial ice samples originating from the Gorner Glacier (Monte Rosa massif, Switzerland) were available for this purpose and show an unexpected modern 39Ar-concentration in some cases. The main result of this thesis, however, is the successful analysis of three depth profiles with eight samples each, taken in the eastern tropical North Atlantic oxygen minimum zone. Due to the reduction of the required sample size by more than two orders of magnitude, it was possible to conduct the sampling with 10 L-Niskin bottles. With the help of the 39Ar-results it is shown, that the transit time distribution of the North Atlantic Deep Water has significantly younger water components than previously assumed. Thus, the outstanding potential of 39Ar for a determination of transit time distributions is demonstrated, which is essential for the understanding of ocean circulations and oxygen supply of the interior of the ocean and which allows, for example, an estimation of the stored anthropogenic carbon.