Elephants were seldom bred in Asian monitored breeding programmes until the last decades, since it was a cheaper and easier method to get more elephants to catch and train wild-cought animals, but there was a regular producation of off spring from females who were kept under semi-domesticated conditions, e.g. were let out in the forest in night-time, were they met wild bulls, and sometimes became pregnant.

Among others, the elephant orphanage in Pinnawela, Sri Lanka, has had great succes, the first baby was born 1984. Thailand has also a large number of captive births, with places like Ayutthaya Elephant camp showing large numbers of new born babies lately.
Burmese working camps may have the largest captive breeding in the world.
Japan had a number of births in Zoos.

Links

Demography of Captive Asian Elephants by Sukumar, R and Krishnamurthy, V and Wemmer, Chris and Rodden, Melissa (1997) Demography of Captive Asian Elephants. In: Zoo Biology, 16 (3). pp. 263-272.