Exemplo de uso do método PUT com a biblioteca do loopJ.

Teste.java

public void callLoopjPut() {

        AsyncHttpClient client = new AsyncHttpClient();

        RequestParams params = new RequestParams();
        params.add("id", "10");
        params.add("name", "Felipe");
        params.add("email", "[email protected]");

        client.put(Teste.this, "http://www.minha-api.com.br/editarCadastro", params, new AsyncHttpResponseHandler() {
            @Override
            public void onSuccess(int statusCode, Header[] headers, byte[] responseBody) {
                // Faz algo
            }

            @Override
            public void onFailure(int statusCode, Header[] headers, byte[] responseBody, Throwable error) {
                // Faz algo
            }
        });
    }