Is it possible to use the old routeProvider in conjunction with the
  stateProvider that ui-router provides?
Short answer
No. Similar structure but different syntax
Long answer
No, but ... You can easily convert ng-view to ui-view a.e. from $routeProvider to $stateProvider.
Consider example ng-view:
$routeProvider  
  .when('/Book/Add', {
    template: '<div class="box" ng-class="classname">Add</div>',
    controller: function($scope) {$scope.classname="add"}
  })
  .when('/Book/Error', {
    templateUrl : 'error.html',
    controller: 'ErrorController'
  })
  .otherwise({redirectTo: '/Book/Error'});
Consider example ui-view:
 $stateProvider
 .state('book', {
                 url: '/Book',
                 abstract: true,
                 templateUrl: 'views/Book.html'
                })
.state('book.add', {
                    url: '/inbox',                    
                    template: '<div class="box" ng-class="classname">Add</div>',
                    controller: function($scope) {$scope.classname="add"}                     
                })
.state('book.error', {
                    url: '/Error',                    
                    templateUrl : 'error.html',
                    controller: 'ErrorController'                   
                })
$urlRouterProvider.otherwise(function ($injector, $location) {
           return '/Book/Error';
          });
Keep in mind that routing syntax will change too.
For example:
 if ($state.current.name === 'login') {
      $state.go('book.add', {});
 }
Instead <a href="/Book/Add">Add</a> we will write <a ui-sref="book.add">Add</a>
And so on ......
As you can see, the syntax is a bit similar. I'm sure you will find a lot of references  about power of $stateProvider. For example https://stackoverflow.com/a/21024270/1631379
Hope I answered on your question