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Neutron-capture reactions play an important role in heavy element nucleosynthesis, since they are the driving force for the two processes that create the vast majority of the heavy elements. When a neutron capture occurs on a short-lived nucleus, it is extremely challenging to study the reaction directly and therefore the use of indirect techniques is essential. The present work reports on such an indirect measurement that provides strong constraints on the 68Ni(n, γ)69Ni reaction rate. This is done by populating the compound nucleus 69Ni via the β decay of 69Co and measuring the γ-ray deexcitation of excited states in 69Ni. The β-Oslo method was used to extract the γ-ray strength function and the nuclear level density. In addition the half-life of 69Co was extracted and found to be in agreement with previous literature values. Before the present results, the 68Ni(n, γ)69Ni reaction was unconstrained and the purely theoretical reaction rate was highly uncertain. The new uncertainty on the reaction rate based on the present experiment (variation between upper and lower limit) is approximately a factor of 3. The commonly used reaction libraries JINA-REACLIB and BRUSLIB are in relatively good agreement with the experimental rate. The impact of the new rate on weak r-process calculations is discussed. [ABSTRACT FROM AUTHOR]