You may remain in the property until you receive a notice to quit. Then at the end of the notice to quit period, you will be required to leave. Your landlord's verbal agreement may not be sufficient to modify the terms of your lease. Most agreements provide that you must put any modifications in writing, otherwise they are not valid. You should continue to pay rent. If you do not pay rent, then your landlord may sue you for the rent owed.
NOTE: This is for information purposes only. No attorney-client relationship is created.
I agree with my colleague - the verbal agreement may not modify the lease. Have you offered to pay an additional deposit and/or pet rent? If you want to stay in the apartment, then you should try and reach an agreement with the landlord - and get it in writing.
My colleagues from out-of-state have correctly answered the question for Colorado. The statute of Frauds requires that all real estate agreements be in writing. Courts will not enforce nor defend the verbal dog agreement, absent some other manifestation, like a confirmatory memo or acceptance of increased rent explicitly for the dog. Leases may be held beyond the 3-day quit or cure notice, however damages accrue. In fact, in certain circumstances, a landlord may seek additional damages for having a dog, even for a second, in a home which is "hypoallergenic" because it causes total damage. If it was advertised as pet free, or hypoallergenic, then be aware that the landlord may terminate and still seek additional damages if you move out on time.