diff --git a/Lib/base64.py b/Lib/base64.py
index e2c597b..adaec1d 100755
--- a/Lib/base64.py
+++ b/Lib/base64.py
@@ -367,10 +367,15 @@
     """
     b = _bytes_from_decode_data(b)
     if adobe:
-        if not (b.startswith(_A85START) and b.endswith(_A85END)):
-            raise ValueError("Ascii85 encoded byte sequences must be bracketed "
-                             "by {!r} and {!r}".format(_A85START, _A85END))
-        b = b[2:-2] # Strip off start/end markers
+        if not b.endswith(_A85END):
+            raise ValueError(
+                "Ascii85 encoded byte sequences must end "
+                "with {!r}".format(_A85END)
+                )
+        if b.startswith(_A85START):
+            b = b[2:-2]  # Strip off start/end markers
+        else:
+            b = b[:-2]
     #
     # We have to go through this stepwise, so as to ignore spaces and handle
     # special short sequences
diff --git a/Lib/test/test_base64.py b/Lib/test/test_base64.py
index 9b853a8..4f86aaa 100644
--- a/Lib/test/test_base64.py
+++ b/Lib/test/test_base64.py
@@ -494,6 +494,7 @@
             eq(base64.a85decode(data, adobe=False), res, data)
             eq(base64.a85decode(data.decode("ascii"), adobe=False), res, data)
             eq(base64.a85decode(b'<~' + data + b'~>', adobe=True), res, data)
+            eq(base64.a85decode(data + b'~>', adobe=True), res, data)
             eq(base64.a85decode('<~%s~>' % data.decode("ascii"), adobe=True),
                res, data)
 
@@ -584,8 +585,6 @@
                                       b"malformed", adobe=True)
         self.assertRaises(ValueError, base64.a85decode,
                                       b"<~still malformed", adobe=True)
-        self.assertRaises(ValueError, base64.a85decode,
-                                      b"also malformed~>", adobe=True)
 
         # With adobe=False (the default), Adobe framing markers are disallowed
         self.assertRaises(ValueError, base64.a85decode,