To migrate the database DB1 from the on-premises Always On availability group (AG1) to the SQL Server instance on the Azure virtual machine (VM1) with minimal downtime, the most effective method is to use a distributed availability group.

A distributed availability group is a special type of availability group that spans two separate availability groups. In this scenario, it will connect the on-premises AG1 with a new availability group on VM1.

1. Preparation: Before you can create the distributed availability group, you must first have a functioning availability group on the target server. Therefore, the necessary preparation step is to create an Always On availability group on VM1. This new AG will act as the secondary (forwarder) in the distributed configuration.
2. Migration: Once the second AG is created on VM1, you can configure a distributed availability group to link it with AG1. This allows for asynchronous or synchronous replication of data from the on-premises primary replica to the Azure VM. The final cutover is a planned failover, which minimizes the downtime required for the migration.

This approach is superior to log shipping for this scenario as it leverages the existing Always On infrastructure and provides a more streamlined failover process. Azure Migrate is a broader service, whereas a distributed availability group is the specific, precise technical feature used for this type of SQL Server migration.

Microsoft Docs, "Distributed availability groups": This document explicitly states, "A distributed availability group is a solution for ... migration." It further explains that a distributed availability group is composed of two distinct availability groups. (See the "Introduction" and "Migration scenarios" sections).

Microsoft Docs, "Configure a distributed availability group": The prerequisites section for configuring a distributed AG clearly states the need for two separate availability groups. Step 1 of the configuration process is to "Create the second availability group" on the cluster that will host the secondary replica, which in this case is the SQL Server instance on VM1.

Microsoft Docs, "Migrate a database using distributed availability groups": This tutorial details the process, confirming that the first step after setting up the target server is to create the secondary availability group, followed by the creation and configuration of the distributed availability group to perform the migration and failover.