Before cardinals start voting for a new pope, they will swear an oath of secrecy in the Sistine Chapel. It will be administered by Cardinal Giovanni Battista Re, the conclave's presiding cardinal. After he reads it, each cardinal elector will touch the Holy Gospels and "promise, pledge and swear" to uphold the oath.
Here's the full text of the oath:
"We, the Cardinal electors present in this election of the Supreme Pontiff promise, pledge and swear, as individuals and as a group, to observe faithfully and scrupulously the prescriptions contained in the Apostolic Constitution of the Supreme Pontiff John Paul II, Universi Dominici Gregis, published on 22 February 1996.
U.S. & World
We likewise promise, pledge and swear that whichever of us by divine disposition is elected Roman Pontiff will commit himself faithfully to carrying out the munus Petrinum of Pastor of the Universal Church and will not fail to affirm and defend strenuously the spiritual and temporal rights and the liberty of the Holy See.
In a particular way, we promise and swear to observe with the greatest fidelity and with all persons, clerical or lay, secrecy regarding everything that in any way relates to the election of the Roman Pontiff and regarding what occurs in the place of the election, directly or indirectly related to the results of the voting; we promise and swear not to break this secret in any way, either during or after the election of the new Pontiff, unless explicit authorization is granted by the same Pontiff; and never to lend support or favor to any interference, opposition or any other form of intervention, whereby secular authorities of whatever order and degree or any group of people or individuals might wish to intervene in the election of the Roman Pontiff."