Is it the battery?.. Whats worng with my phone.?

My phone won't trun on, unless it has the charger connected. And when I displug it, it turns of right away n if I want to trun it back on it has to be charging. Also it say tht the phone is fully charged. Do I have to buy a new battery. And can I do something to not buy a new battery.. So it can work [ if it is the battery]. Whats wrong with my phone.?