I’ve heard that exposure to smoking can cause cleft palate. My husband smokes, and I am worried our baby may suffer. Is this true?

Unfortunately, since. You're a passive smoker, so is your fetus. Tell your hubby, in a gentle but firm way, that prenatal exposure to the > 4, 000 chemicals in environmental tobacco smoke can reduce cortical gray matter & alter devlopment of white matter in your fetus' brain, causing deficits in speech & language skills, visual/perceptual abilities, iq, activity level & behavior, even if he smokes outside.
Cleft and smoking. There is an association between cleft lip/palate and smoking. However it is not know if smoking, or second hand smoke, actually can cause a cleft lip/palate. Smoking is associated with many kinds of birth defects.
No. Cleft lip and palate are birth defects that affect the upper lip and the roof of the mouth. Exposure to smoking for an infant can cause many other physical problems but not cleft palate which is a developmental problem in the womb. I would not expose your infant to cigarette smoke which as i already mentioned can cause serious health issues for your child.
Smoking kills. Babies in households with a smoker have a higher rate of sudden infant death syndrome, even if no one smokes inside the house. Your husband needs to stop smoking.