A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk ( X n ) n ∈ ℕ ∪ { 0 } in a sparse random environment ( S k , λ k ) k ∈ ℤ is a nearest neighbor random walk on ℤ that jumps to the left or to the right with probability 1/2 from every point of ℤ \ { … , S - 1 , S 0 = 0 , S 1 , … } and jumps to the right (left) with the random probability λ k+1 (1 - λ k+1) from the point S k , k ∈ ℤ . Assuming that ( S k - S k - 1 , λ k ) k ∈ ℤ are independent copies of a random vector ( ξ , λ ) ∈ ℕ × ( 0 , 1 ) and the mean E ξ is finite (moderate sparsity) we obtain stable limit laws for X n , properly normalized and centered, as n → ∞. While the case ξ ≤ M a.s. for some deterministic M > 0 (weak sparsity) was analyzed by Matzavinos et al., the case E ξ = ∞ (strong sparsity) will be analyzed in a forthcoming paper.