I have resolved this issue.
In Silverlight, the EntityReference entity has only one constructor, with no parameters. That is why I was assigning values directly to the Id fields' properties. Doing the following instead allowed the Connection to be created:
EntityReference record1Id = new EntityReference();
record1Id.Id = new Guid(_activity_id);
record1Id.LogicalName = _activity_typename;
connToInsert.Record1Id = record1Id;
And so forth for Record2Id, Record1RoleId, and Record2RoleId.