My understanding is that some later ones were transferred to the Royal Alex, yes, but that those would be destroyed every 7 years as per their records management policy, so they may not have any today. But births had to be registered with the province, so Vital Statistics would have some records as well, through its Birth Indexes. Check this link out: http://provincialarchives.alberta.ca/how-to/find-birth-marriage-death-records/births-accordian.aspx They are only publicly searchable up until 1897, it looks like, for privacy reasons. But there are ways to apply to access those records if it’s for you or a direct ancestor/relative (if they’re deceased or can’t apply themselves). If you give them a call or email them, I’m sure they’d be happy to give you some information.
The ones from the earlier period when it was a federally-run hospital would have been sent to Library and Archives Canada in Ottawa, and possibly were also destroyed along the way due to records management policies or other ‘acts of God’. You can contact them for an Access to Information and Privacy (ATIP) request.
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