Hyatt have generated this helpful graphic which explains how their relationship with M life works for properties in Vegas

From today you can book rooms using Hyatt Gold Passport points via the Hyatt Gold Passport web site:

At the Bellagio:

 

On the Bellagio web site there are more room types, but the same price for the Resort Room type, on the date I checked:

 

The full details are, as published before:

•HGP members can now earn (on rates and incidental spend!) and redeem (free nights!) HGP points at 12 iconic MGM Resorts destinations in Las Vegas.
•M life members can earn M life Tier Credits when staying at any of the 450 Hyatt hotels and resorts around the world.
•As an added bonus, guests can earn HGP points and M life Tier Credits simultaneously on eligible spend at these 12 MGM Resorts destinations.
•Eligible stays and nights at the 12 MGM Resorts destinations will count toward Hyatt Gold Passport tier status.
•Coming in August, members of each program will be eligible to opt-in to a matched tier in the other program. Awarded tier match is based on each member’s currently earned status in HGP or M life. Depending on tier, this reciprocal membership may provide a wide variety of on-property benefits, including: ◦Special room offers and upgrades, pre-sale access to premier concert, show and fight tickets, access to exclusive experiences at participating MGM Resorts locations in Las Vegas.
◦Complimentary in-room Internet access, guaranteed room availability, late check-out and more at Hyatt hotels and resorts.

Linking of M life accounts and Hyatt Gold Passport accounts starts next month.