He said one dog allowed and wrote it on the lease. Am I covered?
No, you are not "covered" if it is your dog. The owner of the dog is responsible, not the landlord who simply allows you to have a dog in your apartment.
Dog bite cases can be very expensive and because owners are strictly liable, even minor cases are worth 1000s of dollars. Even if the landlord's insurance covers it because the landlord was in some way negligent the insurance company will come after you as the owner for whatever they paid out.
You are not in a good position. Hopefully, you have renter's insurance or the injuries are very minor or no claim will be made.
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I agree with my colleague - you are not covered. You, as the dog's owner, are liable for any damage caused by your dog. Without knowing the details of the bite it is not possible to advise you, but you should speak with a local attorney.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.