diff --git a/ChangeLog.md b/ChangeLog.md
index b059d22..b32802e 100644
--- a/ChangeLog.md
+++ b/ChangeLog.md
@@ -1,6 +1,12 @@
+## 1.6.0
+
+*   Resolve an internal source of potential flakiness on the bind/close port
+    checks when used in active environments by calling `.shutdown()` before
+    `.close()`.
+
 ## 1.6.0b1
 
-*   Add -h and --help text to the command line tool.
+*   Add `-h` and `--help` text to the command line tool.
 *   The command line interface now defaults to associating the returned port
     with its parent process PID (usually the calling script) when no argument
     was given as that makes more sense.
diff --git a/setup.cfg b/setup.cfg
index 1a5071e..f17d17c 100644
--- a/setup.cfg
+++ b/setup.cfg
@@ -1,7 +1,7 @@
 # https://setuptools.readthedocs.io/en/latest/setuptools.html#configuring-setup-using-setup-cfg-files
 [metadata]
 name = portpicker
-version = 1.6.0b1
+version = 1.6.0
 maintainer = Google LLC
 maintainer_email = greg@krypto.org
 license = Apache 2.0
@@ -27,6 +27,7 @@
     Programming Language :: Python :: 3.9
     Programming Language :: Python :: 3.10
     Programming Language :: Python :: 3.11
+    Programming Language :: Python :: 3.12
     Programming Language :: Python :: Implementation :: CPython
     Programming Language :: Python :: Implementation :: PyPy
 platforms = POSIX, Windows
diff --git a/src/portpicker.py b/src/portpicker.py
index a887781..33805d4 100644
--- a/src/portpicker.py
+++ b/src/portpicker.py
@@ -155,6 +155,14 @@
             return None
         finally:
             if return_socket is None or family != return_family:
+                try:
+                    # Adding this resolved 1 in ~500 flakiness that we were
+                    # seeing from an integration test framework managing a set
+                    # of ports with is_port_free().  close() doesn't move the
+                    # TCP state machine along quickly.
+                    sock.shutdown(socket.SHUT_RDWR)
+                except OSError:
+                    pass
                 sock.close()
         if return_socket is not None and family == return_family:
             return_socket.append(sock)