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Possible Solutions:
1. Expand the cookie domain. The checkout api will see the cookie for the server, so it will probably have a domain of 'www.mystore.com'. Take the cart id and reset it to a wider domain cookie.
The code for that would look something like this (execute this code before changing the location to the checkout page).
document.cookie = 'UltraCartShoppingCartID=' + ultraCart.getCart().cartId + ';domain=.mystore.com;path=/;';
2. Pass the session id as a parameter (version 2.0+ syntax below).
location.href = checkoutPage + '?cartid=' + ultraCart.getCart().cartId;
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