If you want something to switch in less than 60 seconds, then you need either of the following:
Both servers behind the same router, where the router is automatically pinging the primary server and switches the routing tables as soon as the primary goes down. This is generally not feasible in most datacenters unless you're buying a lot of rackspace. Talk to your salesrep.
The DNS entry is configured with a TTL of less than 60 seconds, and a tool on the backup host which monitors the primary, and updates the DNS entry when it detects the primary is inaccessible (and restores it when the primary is accessible again). This will be somewhat unreliable because it's likely that some routers or DNS proxies will ignore a TTL so low (they are usually set at 8 hours or more, not 60 seconds), so some computers or even some whole regions (if the ISP dns is ignoring the low TTL) can wind up using the old IP until the DNS cache clears.