We determine the phase diagram of a one-dimensional Bose gas in the presence of disorder with short- and long-range correlations, the latter decaying with distance as $1/|x|^{1+σ}$. When $σ<0$, the Berezinskii-Kosterlitz-Thouless transition between the superfluid and the localized phase is driven by the long-range correlations and the Luttinger parameter $K$ takes the critical value $K_c(σ)=3/2-σ/2$. The localized phase is a Bose glass for $σ>σ_c=3-π^2/3\simeq -0.289868$, and a Mott glass -- characterized by a vanishing compressibility and a gapless conductivity -- when $σ<σ_c$. Our conclusions, based on the nonperturbative functional renormalization group and perturbative renormalization group, are confirmed by the study of the case $σ=-1$, corresponding to a perfectly correlated disorder in space, where the model is exactly solvable in the semiclassical limit $K\to 0^+$.