The associated production of a Z boson or an off-shell photon $\gamma^*$ with a bottom quark in the forward region is studied using proton-proton collisions at a centre-of-mass energy of $7{\mathrm{\,Te\kern -0.1em V}}$. The Z bosons are reconstructed in the ${\text{Z}/\gamma^*}\rightarrow{\mu^{+}\mu^{-}}$ final state from muons with a transverse momentum larger than $20{\mathrm{\,Ge\kern -0.1em V}}$, while two transverse momentum thresholds are considered for jets ($10{\mathrm{\,Ge\kern -0.1em V}}$ and $20{\mathrm{\,Ge\kern -0.1em V}}$). Both muons and jets are reconstructed in the pseudorapidity range $2.0 < \eta < 4.5$. The results are based on data corresponding to $1.0{\,{fb}^{-1}}$ recorded in 2011 with the LHCb detector. The measurement of the Z+b-jet cross-section is normalized to the Z+jet cross-section. The measured cross-sections are $ \sigma(\text{$\text{Z}/\gamma^*(\mu^{+}\mu^{-})$+b-jet}) = 330 \pm 68 (\text{stat}) \pm 58 (\text{syst}) \pm 12 (\text{lumi}) {\,{fb}}$ for ${$p_{\rm T}$}$(jet)$>10{\mathrm{\,Ge\kern -0.1em V}}$, and $ \sigma(\text{$\text{Z}/\gamma^*(\mu^{+}\mu^{-})$+b-jet}) = 167 \pm 47 (\text{stat}) \pm 29 (\text{syst}) \pm 6 (\text{lumi}) {\,{fb}} $ for ${$p_{\rm T}$}$(jet)$>20{\mathrm{\,Ge\kern -0.1em V}}$.