H1B Visa stamping in new Passport after old passport expiry

I had H1B Visa stamped (valid till August 2016) on my old passport that expired on October 2015. I had renewed my passport on September 2015. I have following 2 questions

a) Should H1B Visa be stamped in my new passport if I travel to India & come back before August 2016, considering that the H1B is valid till Aug 2016 in old passport?

b) If My H1B Validity is extended by another 3 years, Should H1B Visa be stamped in my new passport if I travel to India & come back after August 2016 (old H1B’s expiry) with my new I-797?

  1. No need to get it stamped in new passport. Just carry both passports when traveling in and out of US.

  2. Yes, when current visa stamp expires, any new stamping needs to happen in the new passport.

Thank you Saurabh. It is helpful.