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Let U ħ (g ^) be the untwisted affinization of a symmetrizable quantum Kac-Moody algebra U ħ (g) . For ℓ ∈ C , we construct an ħ -adic quantum vertex algebra V g ^ , ħ (ℓ , 0) , and establish a one-to-one correspondence between ϕ -coordinated V g ^ , ħ (ℓ , 0) -modules and restricted U ħ (g ^) -modules of level ℓ . Suppose that ℓ is a positive integer. We construct a quotient ħ -adic quantum vertex algebra L g ^ , ħ (ℓ , 0) of V g ^ , ħ (ℓ , 0) , and establish a one-to-one correspondence between certain ϕ -coordinated L g ^ , ħ (ℓ , 0) -modules and restricted integrable U ħ (g ^) -modules of level ℓ . Suppose further that g is of finite type. We prove that L g ^ , ħ (ℓ , 0) / ħ L g ^ , ħ (ℓ , 0) is isomorphic to the simple affine vertex algebra L g ^ (ℓ , 0) . [ABSTRACT FROM AUTHOR]