This paper proposes a new algorithm for the generation of unambiguous Büchi automata (UBA) from LTL formulas. Unlike existing tableau-based LTL-to-UBA translations, our algorithm deals with very weak alternating automata (VWAA) as an intermediate representation. It relies on a new notion of unambiguity for VWAA and a disambiguation procedure for VWAA. We introduce optimizations on the VWAA level and new LTL simplifications targeted at generating small UBA. We report on an implementation of the construction in our tool Duggi and discuss experimental results that compare the automata sizes and computation times of Duggi with the tableau-based LTL-to-UBA translation of the SPOT tool set. Our experiments also cover the analysis of Markov chains under LTL specifications, which is an important application of UBA.