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We show how to construct an existentially unforgeable secure signature scheme from any scheme satisfies only a weak notion of security in the standard model. This construction method combines a weakly secure signature and a one-time signature. However, key generation of the resulted fully secure signature is the same as the key generation of weak signature. Therefore the length of the public key in our fully secure signature is independent of that of the one-time signature. Our conversion from a weakly secure signature scheme to an existentially unforgeable secure signature scheme is simple, efficient and provably secure in the standard model (that is, security of the resulting scheme does not rely on the random oracle model). Our results yield a new construction of existentially unforgeable secure signature in the standard model. Furthermore, we show two efficient instantiations without random oracles converted from two previous weakly secure signature schemes.