Not all apologies are equal. English writer Gilbert K. Chesterton said that "a stiff apology is a second insult . . . The injured party does not want to be compensated because he has been wronged; he wants to be healed because he has been hurt." Simply apologizing doesn't change what happened, but a good apology can go a long way toward repairing emotional damage.
Acknowledge what you did wrong -- to yourself and to your boyfriend. Come clean about the promise you broke and anything hurtful you did involving that promise. Don't leave out information you might have to apologize for later. Tell your boyfriend that you know breaking the promise was hurtful and violated his trust.
Take responsibility. When you say you are sorry, don't add a "but." Don't say "I'm sorry you're upset that I broke my promise," because that takes the responsibility off you and puts it on him. Although you might offer an explanation for your actions, this can be a slippery slope into rationalization, which is another way to avoid responsibility.
Give your boyfriend time to express his feelings. He might be angry or ask questions. Listen and respond empathetically. Although you may be tempted to relate every extenuating circumstance that led to breaking your promise, don't. It's more hurtful to hear all the reasons why something is not one's fault than to simply hear, "I'm so sorry. I broke a promise and I was wrong."
Make amends. Ask your boyfriend if there is anything you can do to help repair the damage (there may not be). Your boyfriend may be hurt, mad or angry, and you will likely need to give him time to recover. Although you can't take back what you did, you can work to prove you're trustworthy, particularly by not repeating the offense.