diff --git a/changelog.d/1671.change.rst b/changelog.d/1671.change.rst
new file mode 100644
index 0000000000..95ae49da64
--- /dev/null
+++ b/changelog.d/1671.change.rst
@@ -0,0 +1 @@
+Fixed issue with the PEP 517 backend that prevented building a wheel when the ``dist/`` directory contained existing ``.whl`` files.
diff --git a/setuptools/build_meta.py b/setuptools/build_meta.py
index 47cbcbf684..e40904a5df 100644
--- a/setuptools/build_meta.py
+++ b/setuptools/build_meta.py
@@ -35,6 +35,7 @@
 
 import setuptools
 import distutils
+from setuptools.py31compat import TemporaryDirectory
 
 from pkg_resources import parse_requirements
 
@@ -182,14 +183,22 @@ def build_wheel(self, wheel_directory, config_settings=None,
                     metadata_directory=None):
         config_settings = self._fix_config(config_settings)
         wheel_directory = os.path.abspath(wheel_directory)
-        sys.argv = sys.argv[:1] + ['bdist_wheel'] + \
-            config_settings["--global-option"]
-        self.run_setup()
-        if wheel_directory != 'dist':
-            shutil.rmtree(wheel_directory)
-            shutil.copytree('dist', wheel_directory)
 
-        return _file_with_extension(wheel_directory, '.whl')
+        # Build the wheel in a temporary directory, then copy to the target
+        with TemporaryDirectory(dir=wheel_directory) as tmp_dist_dir:
+            sys.argv = (sys.argv[:1] +
+                        ['bdist_wheel', '--dist-dir', tmp_dist_dir] +
+                        config_settings["--global-option"])
+            self.run_setup()
+
+            wheel_basename = _file_with_extension(tmp_dist_dir, '.whl')
+            wheel_path = os.path.join(wheel_directory, wheel_basename)
+            if os.path.exists(wheel_path):
+                # os.rename will fail overwriting on non-unix env
+                os.remove(wheel_path)
+            os.rename(os.path.join(tmp_dist_dir, wheel_basename), wheel_path)
+
+        return wheel_basename
 
     def build_sdist(self, sdist_directory, config_settings=None):
         config_settings = self._fix_config(config_settings)
diff --git a/setuptools/py31compat.py b/setuptools/py31compat.py
index 1a0705ece3..e1da7ee2a2 100644
--- a/setuptools/py31compat.py
+++ b/setuptools/py31compat.py
@@ -17,9 +17,9 @@ class TemporaryDirectory:
         errors on deletion.
         """
 
-        def __init__(self):
+        def __init__(self, **kwargs):
             self.name = None  # Handle mkdtemp raising an exception
-            self.name = tempfile.mkdtemp()
+            self.name = tempfile.mkdtemp(**kwargs)
 
         def __enter__(self):
             return self.name
diff --git a/setuptools/tests/test_build_meta.py b/setuptools/tests/test_build_meta.py
index 90400afc50..88e29ffe1a 100644
--- a/setuptools/tests/test_build_meta.py
+++ b/setuptools/tests/test_build_meta.py
@@ -157,7 +157,6 @@ def test_build_wheel(self, build_backend):
 
         assert os.path.isfile(os.path.join(dist_dir, wheel_name))
 
-    @pytest.mark.xfail(reason="Known error, see GH #1671")
     def test_build_wheel_with_existing_wheel_file_present(self, tmpdir_cwd):
         # Building a wheel should still succeed if there's already a wheel
         # in the wheel directory
@@ -195,6 +194,12 @@ def test_build_wheel_with_existing_wheel_file_present(self, tmpdir_cwd):
         assert os.path.isfile(os.path.join(dist_dir, wheel_one))
         assert wheel_one != wheel_two
 
+        # and if rebuilding the same wheel?
+        open(os.path.join(dist_dir, wheel_two), 'w').close()
+        wheel_three = self.get_build_backend().build_wheel(dist_dir)
+        assert wheel_three == wheel_two
+        assert os.path.getsize(os.path.join(dist_dir, wheel_three)) > 0
+
     def test_build_sdist(self, build_backend):
         dist_dir = os.path.abspath('pip-sdist')
         os.makedirs(dist_dir)