We consider the problem of secure distributed data storage under the paradigm of "weak security", in which no "meaningful" information is leaked to the eavesdropper. More specifically, the eavesdropper cannot get any information about any individual data packet or a small group of packets. The key benefit of the weak security paradigm is that it incurs no loss in the storage capacity, which makes it practically appealing. In this work, we present the constructions of outer codes to weakly secure the Product-Matrix based regenerating codes (proposed by Rashmi et al.), when the eavesdropper can observe only the stored data (and not the downloaded data). We show that the proposed codes have good security properties and are easy to implement in practical settings.