The well-known phrase "first, do no harm" is not in the Hippocratic Oath.
The context of the phrase comes from The History of Epidemics, which is part of the Hippocratic corpus:
"...But in those cases in which there was no danger, and where a concoction was made in proper time, it should be considered whether the translations of humours were good or critical signs. Concoctions always portend a crisis, and safety from the disease; but crudities, or inconcoctions are soon converted into bad translations, or a defect or want of crisis, or pain, or a duration of the disease, or death, or a relapse. What will happen from these should be considered from other circumstances, as to know what is past, to discern what is present, and to predict the future. And these two things in disease are particularly to be attended to, to do good, and not to do harm. The whole art of medicine may be circumscribed in three distinctions, medicine, the sick-man, and the physician who is the minister of the art; and the conflict lies between the sick-person, the physician, and the disease."
I swear by Apollo Physician and Asclepius and Hygieia and Panaceia and all the gods and goddesses, making them my witnesses, that I will fulfil according to my ability and judgment this oath and this covenant:
To hold him who has taught me this art as equal to my parents and to live my life in partnership with him, and if he is in need of money to give him a share of mine, and to regard his offspring as equal to my brothers in male lineage and to teach them this art - if they desire to learn it - without fee and covenant; to give a share of precepts and oral instruction and all the other learning to my sons and to the sons of him who has instructed me and to pupils who have signed the covenant and have taken an oath according to the medical law, but no one else.
I will apply dietetic measures for the benefit of the sick according to my ability and judgment; I will keep them from harm and injustice.
I will neither give a deadly drug to anybody who asked for it, nor will I make a suggestion to this effect. Similarly I will not give to a woman an abortive remedy. In purity and holiness I will guard my life and my art.
I will not use the knife, not even on sufferers from stone, but will withdraw in favor of such men as are engaged in this work.
Whatever houses I may visit, I will come for the benefit of the sick, remaining free of all intentional injustice, of all mischief and in particular of sexual relations with both female and male persons, be they free or slaves.
What I may see or hear in the course of the treatment or even outside of the treatment in regard to the life of men, which on no account one must spread abroad, I will keep to myself, holding such things shameful to be spoken about.
If I fulfil this oath and do not violate it, may it be granted to me to enjoy life and art, being honored with fame among all men for all time to come; if I transgress it and swear falsely, may the opposite of all this be my lot.
Translation from the Greek by Ludwig Edelstein. From The Hippocratic Oath: Text, Translation, and Interpretation, by Ludwig Edelstein. Baltimore: The Johns Hopkins Press, 1943.