의원의 = 의원(n) + 의(possessive particle) = of a/the doctor.
의원의 말 = the word of a/the doctor; what the doctor said.
의원 = old term for "doctor". It has largely been replaced by the newer term 의사.
의 is one of the trickiest vowels with three accepted pronunciations.
1) at the beginning of a noun (e.g. 의사, 의심): [으이] - smoothly transitioning with short and light 이.
2) at the end of a noun (정의, 심의). Either [으이] like #1 or simple [이] ([이] is officially accepted).
3) possessive particle (병원의, 선생님의). Degenerates to 에- or 애-like sound.
Note that the possessive 의 is more often dropped in speech than actually pronounced. For example, in casual conversation, people might say 의원 말 instead of 의원의 말. When 의 is pronounced, it will sound like 의원에 말 or 의원애 말.