Search for the decay of nature's rarest isotope Tam180

^Tam₁₈₀ is the rarest naturally occurring quasistable isotope and the longest lived metastable state which is known. Its possible decay via the β^- or the electron capture channel has never been observed. This article presents a search for the decay of ^Tam₁₈₀ with an ultralow background Sandwich HPGe γ spectrometry setup in the HADES underground laboratory. No signal is observed and improved lower partial half-life limits are set with a Bayesian analysis to 5.8 ×10¹⁶yr for the β^- channel and 2.0 ×10¹⁷yr for the electron capture channel (90 % credibility). The total half-life of ^Tam₁₈₀ is longer than 4.5 ×10¹⁶yr . This is more than a factor of two improvement compared to previous searches.