subtly convincing her steward, Antonio, to marry her.

Antonio doesn't take much convincing to marry the Duchess because he's already in love with her, but he's never acted on his feelings towards her because of his servant-class status.

The Duchess dismisses Antonio's concerns by convincing him that whether he's low-born or high-born, he's still a "complete man" and no less worthy of her than any other man.

DUCHESS. You were ill to sell yourself.
This darkening of your worth is not like that
Which tradesmen use i’th’ city; their false lights
Are to rid bad wares off. And I must tell you,
If you will know where breathes a complete man
(I speak it without flattery) turn your eyes,
And progress through yourself (1.3).

A few lines earlier in the scene, after she puts a ring on his finger while he kneels before her, the Duchess offers to raise Antonio up from his lowly birth as she raises him up from the floor.

DUCHESS. Sir, this goodly roof of yours, is too low built;
I cannot stand upright in’t nor discourse,
Without I raise it higher. Raise yourself,
Or, if you please, my hand to help you: so.

[Raises him.] (1.3.)

Ever mindful of his own place in the world, Antonio responds that being raised up imposes certain responsibilities and certain afflictions, such as ambition, which he had not previously encountered in his lower state in life.

ANTONIO. Ambition, madam, is a great man’s madness,
That is not kept in chains and close-pent rooms
But in fair lightsome lodgings and is girt
With the wild noise of prattling visitants
Which makes it lunatic beyond all cure.
Conceive not I am so stupid but I aim
Whereto your favors tend: but he’s a fool,
That being a-cold, would thrust his hands i’th’ fire
To warm them (1.3.).

Antonio is concerned that being made "great" by marrying the Duchess will cause him to suffer the "madness" of all-consuming ambition that accompanies status, power, and wealth, and which ambition will ultimately destroy him, like thrusting a cold hand into a fire.

Antonio is overcome by the speed with which this is taking place. He's also taken aback by the appearance of Cariola, the Duchess's lady-in-waiting and confidant who's been eavesdropping on the scene in order to verify the vows of marriage that the Duchess and Antonio make between themselves.

The scene ends happily, with Antonio and the Duchess going off together to their marriage bed, but Cariola casts a shadow over their marriage and their happiness with her last lines in the scene:

CARIOLA. Whether the spirit of greatness or of woman
Reign most in her, I know not, but it shows
A fearful madness. I owe her much of pity (1.3).

Antonio is a humble and straight man—or at least he appears to be so among an incredibly crooked court. He speaks this line in Act I, Scene 3 to the Duchess, with whom he is having a conversation heavy with implication and innuendo. For example, the Duchess says that one of Antonio's eyes are bloodshot, and she gives him her ring for its "healing properties." This is of course her wedding ring, and she is implying that Antonio should marry her.

She goes on to say that his head is too low, and she needs to "raise him up" to talk to him. She means this in terms of court status, not literally. It is then that he says "Ambition, madam, is a great man's madness." Antonio is ambivalent in regard to pursuing higher things and thinks none of this would be worth the trouble and scandal it would cause. He fears the intoxicating effects of power and how they may change him from his righteous ways.

The line "Ambition, madam, is a great man's madness" is spoken by Antonio in act I, scene iii. It shows Antonio's ambivalence at the prospect of marrying the Duchess. Antonio is merely a humble steward, the man who runs the Duchess's household. He knows that marriage between an aristocratic lady and one of her servants would cause a huge scandal in such a class-conscious society.

Furthermore, Antonio is concerned that being elevated to such a high position in life will tempt him to become ambitious, and that way madness lies. (Just one look at some of the other characters at court tells us that Antonio is absolutely right to be worried.) Antonio is a fundamentally decent man, a man of great integrity who's generally happy with his lot in life. Although marrying the Duchess will give Antonio an entree into a world of wealth, glamor, and high social status, it could also just as easily corrupt his soul, making him ambitious for riches and power. Antonio senses the dangers of getting married to the Duchess, and so he tries to talk her out of it.