We consider discrete memoryless channels with input and output alphabet size n whose channel transition matrix consists of entries that are independent and identically distributed according to some probability distribution v on (R≥0, B(R≥0)) before being normalized, where v is such that E[X log X)2 1 1 := E[X] and μ 2 := E[X log X] for a random variable X with distribution v. We prove that in the limit as n → ∞, the capacity of such a channel converges to μ 2 /μ 1 − log μ 1 almost surely and in L2. We further show that the capacity of these random channels converges to this asymptotic value exponentially in n.