Get the hash from the signature. Details can be found here: http://stackoverflow.com/questions/22506131/get-hash-of-java-security-signature-create-signature-of-given-hash

SignatureTest.java

import java.security.KeyPair;
import java.security.KeyPairGenerator;
import java.security.Signature;

import javax.crypto.Cipher;

public class SignatureTest {

    public static void main(String[] args) throws Exception {
        /**
         * Create two keypairs and a dummy data object which we sign
         */
        KeyPairGenerator keyGen = KeyPairGenerator.getInstance("RSA");
        keyGen.initialize(512);
        KeyPair keyPair1 = keyGen.generateKeyPair();
        KeyPair keyPair2 = keyGen.generateKeyPair();
        String data = new String("dummy");

        /**
         * Sign with keypair 1
         */
        Signature signature = Signature.getInstance("SHA1withRSA");
        signature.initSign(keyPair1.getPrivate());
        signature.update(data.getBytes());
        byte[] signature1 = signature.sign();

        /**
         * Decrypt signature to get hash of the object
         */
        Cipher rsa = Cipher.getInstance("RSA");
        rsa.init(Cipher.DECRYPT_MODE, keyPair1.getPublic());
        byte[] hash = rsa.doFinal(signature1);

        /**
         * Encrypt with keypair 2 to get the new signature
         */
        rsa = Cipher.getInstance("RSA");
        rsa.init(Cipher.ENCRYPT_MODE, keyPair2.getPrivate());
        byte[] signature2 = rsa.doFinal(hash);

        /**
         * Verify with another signature instance
         */
        signature = Signature.getInstance("SHA1withRSA");
        signature.initVerify(keyPair2.getPublic());
        signature.update(data.getBytes());
        System.out.println("Verified new key: " + signature.verify(signature2));
    }
}