diff --git a/src/io/copy.rs b/src/io/copy.rs
index f05ed0e1..e8d5d733 100644
--- a/src/io/copy.rs
+++ b/src/io/copy.rs
@@ -7,6 +7,12 @@ use crate::io::{self, BufRead, BufReader, Read, Write};
 use crate::task::{Context, Poll};
 use crate::utils::Context as _;
 
+// Note: There are two otherwise-identical implementations of this
+// function because unstable has removed the `?Sized` bound for the
+// reader and writer and accepts `R` and `W` instead of `&mut R` and
+// `&mut W`. If making a change to either of the implementations,
+// ensure that you copy it into the other.
+
 /// Copies the entire contents of a reader into a writer.
 ///
 /// This function will continuously read data from `reader` and then
@@ -57,6 +63,7 @@ where
             #[pin]
             writer: W,
             amt: u64,
+            reader_eof: bool
         }
     }
 
@@ -69,13 +76,20 @@ where
 
         fn poll(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Self::Output> {
             let mut this = self.project();
+
             loop {
-                let buffer = futures_core::ready!(this.reader.as_mut().poll_fill_buf(cx))?;
-                if buffer.is_empty() {
+                if *this.reader_eof {
                     futures_core::ready!(this.writer.as_mut().poll_flush(cx))?;
                     return Poll::Ready(Ok(*this.amt));
                 }
 
+                let buffer = futures_core::ready!(this.reader.as_mut().poll_fill_buf(cx))?;
+
+                if buffer.is_empty() {
+                    *this.reader_eof = true;
+                    continue;
+                }
+
                 let i = futures_core::ready!(this.writer.as_mut().poll_write(cx, buffer))?;
                 if i == 0 {
                     return Poll::Ready(Err(io::ErrorKind::WriteZero.into()));
@@ -89,6 +103,7 @@ where
     let future = CopyFuture {
         reader: BufReader::new(reader),
         writer,
+        reader_eof: false,
         amt: 0,
     };
     future.await.context(|| String::from("io::copy failed"))
@@ -144,6 +159,7 @@ where
             #[pin]
             writer: W,
             amt: u64,
+            reader_eof: bool
         }
     }
 
@@ -156,13 +172,20 @@ where
 
         fn poll(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Self::Output> {
             let mut this = self.project();
+
             loop {
-                let buffer = futures_core::ready!(this.reader.as_mut().poll_fill_buf(cx))?;
-                if buffer.is_empty() {
+                if *this.reader_eof {
                     futures_core::ready!(this.writer.as_mut().poll_flush(cx))?;
                     return Poll::Ready(Ok(*this.amt));
                 }
 
+                let buffer = futures_core::ready!(this.reader.as_mut().poll_fill_buf(cx))?;
+
+                if buffer.is_empty() {
+                    *this.reader_eof = true;
+                    continue;
+                }
+
                 let i = futures_core::ready!(this.writer.as_mut().poll_write(cx, buffer))?;
                 if i == 0 {
                     return Poll::Ready(Err(io::ErrorKind::WriteZero.into()));
@@ -176,6 +199,7 @@ where
     let future = CopyFuture {
         reader: BufReader::new(reader),
         writer,
+        reader_eof: false,
         amt: 0,
     };
     future.await.context(|| String::from("io::copy failed"))
diff --git a/tests/io_copy.rs b/tests/io_copy.rs
new file mode 100644
index 00000000..394c34f8
--- /dev/null
+++ b/tests/io_copy.rs
@@ -0,0 +1,72 @@
+use std::{
+    io::Result,
+    pin::Pin,
+    task::{Context, Poll},
+};
+
+struct ReaderThatPanicsAfterEof {
+    read_count: usize,
+    has_sent_eof: bool,
+    max_read: usize,
+}
+
+impl async_std::io::Read for ReaderThatPanicsAfterEof {
+    fn poll_read(
+        mut self: Pin<&mut Self>,
+        _cx: &mut Context<'_>,
+        buf: &mut [u8],
+    ) -> Poll<Result<usize>> {
+        if self.has_sent_eof {
+            panic!("this should be unreachable because we should not poll after eof (Ready(Ok(0)))")
+        } else if self.read_count >= self.max_read {
+            self.has_sent_eof = true;
+            Poll::Ready(Ok(0))
+        } else {
+            self.read_count += 1;
+            Poll::Ready(Ok(buf.len()))
+        }
+    }
+}
+
+struct WriterThatTakesAWhileToFlush {
+    max_flush: usize,
+    flush_count: usize,
+}
+
+impl async_std::io::Write for WriterThatTakesAWhileToFlush {
+    fn poll_write(self: Pin<&mut Self>, _cx: &mut Context<'_>, buf: &[u8]) -> Poll<Result<usize>> {
+        Poll::Ready(Ok(buf.len()))
+    }
+
+    fn poll_flush(mut self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Result<()>> {
+        self.flush_count += 1;
+        if self.flush_count >= self.max_flush {
+            Poll::Ready(Ok(()))
+        } else {
+            cx.waker().wake_by_ref();
+            Poll::Pending
+        }
+    }
+
+    fn poll_close(self: Pin<&mut Self>, _cx: &mut Context<'_>) -> Poll<Result<()>> {
+        Poll::Ready(Ok(()))
+    }
+}
+
+#[test]
+fn io_copy_does_not_poll_after_eof() {
+    async_std::task::block_on(async {
+        let mut reader = ReaderThatPanicsAfterEof {
+            has_sent_eof: false,
+            max_read: 10,
+            read_count: 0,
+        };
+
+        let mut writer = WriterThatTakesAWhileToFlush {
+            flush_count: 0,
+            max_flush: 10,
+        };
+
+        assert!(async_std::io::copy(&mut reader, &mut writer).await.is_ok());
+    })
+}