Artin (1923) wrote that Dedekind proved the case of this conjecture concerning pure cubic number fields. Dedekind published his article in 1900, but writes that it is a reworking of a draft he had written in 1871/72. The conjecture was named after Dedekind by van der Waall in 1974.
Despite Artin's claim, the integrality of the quotient $\zeta_K(s)/\zeta(s)$ is not addressed at all by Dedekind (Hecke proved that $\zeta_K(s)$ extends to a meromorphic function on the complex plane in 1917). Dedekind proved that $\zeta_K(s)/\zeta(s)$ for pure cubic extensions may be written as a linear combination of Epstein zeta functions. Epstein proved in 1903 that these are meromorphic with simple poles in $s = 1$, and Artin realized in 1923 that this implies that the quotient is entire - I guess this explains his remark.
In any case Dedekind had nothing to do with this conjecture.
Edit. Landau [Über die Wurzeln der Zetafunktion eines algebraischen Zahlkörpers, Math. Ann. 79 (1919), 388-401] writes on p. 390: "I have to be careful with its formulation since it is not known whether $\zeta_k(s)/\zeta(s)$ is always an entire function".