Designing approximate all-pairs distance oracles in the fully dynamic setting is one of the central problems in dynamic graph algorithms. Despite extensive research on this topic, the first result breaking the $O(\sqrt{n})$ barrier on the update time for any non-trivial approximation was introduced only recently by Forster, Goranci and Henzinger [SODA'21] who achieved $m^{1/ρ+o(1)}$ amortized update time with a $O(\log n)^{3ρ-2}$ factor in the approximation ratio, for any parameter $ρ\geq 1$. In this paper, we give the first constant-stretch fully dynamic distance oracle with a small polynomial update and query time. Prior work required either at least a poly-logarithmic approximation or much larger update time. Our result gives a more fine-grained trade-off between stretch and update time, for instance we can achieve constant stretch of $O(\frac{1}{ρ^2})^{4/ρ}$ in amortized update time $\tilde{O}(n^ρ)$, and query time $\tilde{O}(n^{ρ/8})$ for a constant parameter $ρ<1$. Our algorithm is randomized and assumes an oblivious adversary. A core technical idea underlying our construction is to design a black-box reduction from decremental approximate hub-labeling schemes to fully dynamic distance oracles, which may be of independent interest. We then apply this reduction repeatedly to an existing decremental algorithm to bootstrap our fully dynamic solution.