Algebraic Manipulation Detection (AMD) Codes detect adversarial noise that is added to a coded message and stored in a storage that is opaque to the adversary. We study AMD codes when the storage can leak up to ρ\log|G| bits of information about the stored codeword, where G is the group in which the stored codeword lives and ρis a constant. We propose ρ-AMD codes that provide protection in this new setting, and define weak and strong ρ-AMD codes that provide security for a random and an arbitrary message, respectively. We derive concrete and asymptotic bounds for the efficiency of these codes featuring a rate upper bound of 1-ρfor the strong codes. We also define the class of ρ^{LV}-AMD codes that provide protection when leakage is in the form of a number of codeword components, and give constructions featuring a strong ρ^{LV}-AMD codes that asymptotically achieve the rate 1-ρ. We describe applications of ρ-AMD codes to, (i) robust ramp secret sharing scheme and, (ii) wiretap II channel when the adversary can eavesdrop a ρfraction of codeword components and tamper with all components of the codeword.