diff --git a/Doc/library/asyncio-stream.rst b/Doc/library/asyncio-stream.rst
index ba534f9903fb49..72355d356f2052 100644
--- a/Doc/library/asyncio-stream.rst
+++ b/Doc/library/asyncio-stream.rst
@@ -295,6 +295,24 @@ StreamWriter
       be resumed.  When there is nothing to wait for, the :meth:`drain`
       returns immediately.
 
+   .. coroutinemethod:: start_tls(sslcontext, \*, server_hostname=None, \
+                          ssl_handshake_timeout=None)
+
+      Upgrade an existing stream-based connection to TLS.
+
+      Parameters:
+
+      * *sslcontext*: a configured instance of :class:`~ssl.SSLContext`.
+
+      * *server_hostname*: sets or overrides the host name that the target
+        server's certificate will be matched against.
+
+      * *ssl_handshake_timeout* is the time in seconds to wait for the TLS
+        handshake to complete before aborting the connection.  ``60.0`` seconds
+        if ``None`` (default).
+
+      .. versionadded:: 3.8
+
    .. method:: is_closing()
 
       Return ``True`` if the stream is closed or in the process of
diff --git a/Doc/whatsnew/3.11.rst b/Doc/whatsnew/3.11.rst
index df0b0a7fbebec9..3fa64df1b8bfad 100644
--- a/Doc/whatsnew/3.11.rst
+++ b/Doc/whatsnew/3.11.rst
@@ -246,6 +246,10 @@ asyncio
   :meth:`~asyncio.AbstractEventLoop.sock_recvfrom_into`.
   (Contributed by Alex Grönholm in :issue:`46805`.)
 
+* Add :meth:`~asyncio.streams.StreamWriter.start_tls` method for upgrading
+  existing stream-based connections to TLS. (Contributed by Ian Good in
+  :issue:`34975`.)
+
 fractions
 ---------
 
diff --git a/Lib/asyncio/streams.py b/Lib/asyncio/streams.py
index 080d8a62cde1e2..a568c4e4b295f0 100644
--- a/Lib/asyncio/streams.py
+++ b/Lib/asyncio/streams.py
@@ -217,6 +217,13 @@ def _stream_reader(self):
             return None
         return self._stream_reader_wr()
 
+    def _replace_writer(self, writer):
+        loop = self._loop
+        transport = writer.transport
+        self._stream_writer = writer
+        self._transport = transport
+        self._over_ssl = transport.get_extra_info('sslcontext') is not None
+
     def connection_made(self, transport):
         if self._reject_connection:
             context = {
@@ -371,6 +378,20 @@ async def drain(self):
             await sleep(0)
         await self._protocol._drain_helper()
 
+    async def start_tls(self, sslcontext, *,
+                        server_hostname=None,
+                        ssl_handshake_timeout=None):
+        """Upgrade an existing stream-based connection to TLS."""
+        server_side = self._protocol._client_connected_cb is not None
+        protocol = self._protocol
+        await self.drain()
+        new_transport = await self._loop.start_tls(  # type: ignore
+            self._transport, protocol, sslcontext,
+            server_side=server_side, server_hostname=server_hostname,
+            ssl_handshake_timeout=ssl_handshake_timeout)
+        self._transport = new_transport
+        protocol._replace_writer(self)
+
 
 class StreamReader:
 
diff --git a/Lib/test/test_asyncio/test_streams.py b/Lib/test/test_asyncio/test_streams.py
index 227b2279e172c8..a7d17894e1c526 100644
--- a/Lib/test/test_asyncio/test_streams.py
+++ b/Lib/test/test_asyncio/test_streams.py
@@ -706,6 +706,69 @@ async def client(path):
 
         self.assertEqual(messages, [])
 
+    @unittest.skipIf(ssl is None, 'No ssl module')
+    def test_start_tls(self):
+
+        class MyServer:
+
+            def __init__(self, loop):
+                self.server = None
+                self.loop = loop
+
+            async def handle_client(self, client_reader, client_writer):
+                data1 = await client_reader.readline()
+                client_writer.write(data1)
+                await client_writer.drain()
+                assert client_writer.get_extra_info('sslcontext') is None
+                await client_writer.start_tls(
+                    test_utils.simple_server_sslcontext())
+                assert client_writer.get_extra_info('sslcontext') is not None
+                data2 = await client_reader.readline()
+                client_writer.write(data2)
+                await client_writer.drain()
+                client_writer.close()
+                await client_writer.wait_closed()
+
+            def start(self):
+                sock = socket.create_server(('127.0.0.1', 0))
+                self.server = self.loop.run_until_complete(
+                    asyncio.start_server(self.handle_client,
+                                         sock=sock))
+                return sock.getsockname()
+
+            def stop(self):
+                if self.server is not None:
+                    self.server.close()
+                    self.loop.run_until_complete(self.server.wait_closed())
+                    self.server = None
+
+        async def client(addr):
+            reader, writer = await asyncio.open_connection(*addr)
+            writer.write(b"hello world 1!\n")
+            await writer.drain()
+            msgback1 = await reader.readline()
+            assert writer.get_extra_info('sslcontext') is None
+            await writer.start_tls(test_utils.simple_client_sslcontext())
+            assert writer.get_extra_info('sslcontext') is not None
+            writer.write(b"hello world 2!\n")
+            await writer.drain()
+            msgback2 = await reader.readline()
+            writer.close()
+            await writer.wait_closed()
+            return msgback1, msgback2
+
+        messages = []
+        self.loop.set_exception_handler(lambda loop, ctx: messages.append(ctx))
+
+        server = MyServer(self.loop)
+        addr = server.start()
+        msg1, msg2 = self.loop.run_until_complete(client(addr))
+        server.stop()
+
+        self.assertEqual(messages, [])
+        self.assertEqual(msg1, b"hello world 1!\n")
+        self.assertEqual(msg2, b"hello world 2!\n")
+
     @unittest.skipIf(sys.platform == 'win32', "Don't have pipes")
     def test_read_all_from_pipe_reader(self):
         # See asyncio issue 168.  This test is derived from the example
diff --git a/Misc/NEWS.d/next/Library/2019-05-06-23-36-34.bpo-34975.eb49jr.rst b/Misc/NEWS.d/next/Library/2019-05-06-23-36-34.bpo-34975.eb49jr.rst
new file mode 100644
index 00000000000000..1576269da99ee0
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2019-05-06-23-36-34.bpo-34975.eb49jr.rst
@@ -0,0 +1,3 @@
+Adds a ``start_tls()`` method to :class:`~asyncio.streams.StreamWriter`,
+which upgrades the connection with TLS using the given
+:class:`~ssl.SSLContext`.