diff --git a/conn.go b/conn.go
index 229c380c5..1e60af2d8 100644
--- a/conn.go
+++ b/conn.go
@@ -359,8 +359,13 @@ func (c *Conn) Prepare(ctx context.Context, name, sql string) (sd *pgconn.Statem
 	sd, err = c.pgConn.Prepare(ctx, psName, sql, nil)
 	if err != nil {
 		var pErr *pgconn.PrepareError
-		if errors.As(err, &pErr) {
-			c.failedDescribeStatement = psKey
+		if errors.As(err, &pErr) && pErr.ParseComplete {
+			// The server-side statement was created under psName — the name sent in
+			// Parse. In the name == sql case psKey is the SQL text, and deallocating
+			// by it would close a nonexistent statement while leaking the real one.
+			// When Parse never completed no statement was created at all, so there
+			// is nothing to clean up.
+			c.failedDescribeStatement = psName
 		}
 		return nil, err
 	}
diff --git a/conn_test.go b/conn_test.go
index db3578b4b..60fd63e88 100644
--- a/conn_test.go
+++ b/conn_test.go
@@ -3,7 +3,9 @@ package pgx_test
 import (
 	"bytes"
 	"context"
+	"crypto/sha256"
 	"database/sql"
+	"encoding/hex"
 	"io"
 	"net"
 	"os"
@@ -532,6 +534,129 @@ func TestPrepareHandlesTimeoutBetweenParseAndDescribe(t *testing.T) {
 	require.NotNil(t, psd)
 }
 
+// https://github.com/jackc/pgx/issues/2640
+func TestPrepareWithDigestedNameHandlesTimeoutBetweenParseAndDescribe(t *testing.T) {
+	// Not parallel because it is a timing sensitive test.
+	//
+	// stdlib (and therefore database/sql) calls Prepare(ctx, sql, sql). In that case the statement is prepared on the
+	// server under stmt_<digest> while the client keys it by the SQL text. Cleanup after a Describe phase failure must
+	// deallocate the digest name — deallocating by the SQL text closes a nonexistent statement, so the leaked
+	// statement stays on the server and re-preparing the same SQL fails with 42P05 for the rest of the connection's
+	// life.
+
+	config, err := pgx.ParseConfig(os.Getenv("PGX_TEST_DATABASE"))
+	require.NoError(t, err)
+
+	var faultyConn *faultyconn.Conn
+	config.AfterNetConnect = func(ctx context.Context, config *pgconn.Config, conn net.Conn) (net.Conn, error) {
+		faultyConn = faultyconn.New(conn)
+		return faultyConn, nil
+	}
+
+	ctx := context.Background()
+	conn, err := pgx.ConnectConfig(ctx, config)
+	require.NoError(t, err)
+	defer closeConn(t, conn)
+	require.NotNil(t, faultyConn)
+
+	pgxtest.SkipCockroachDB(t, conn, "Induced error does not occur on CockroachDB")
+
+	_, err = conn.Exec(ctx, "set statement_timeout = '100ms'")
+	require.NoError(t, err)
+
+	faultyConn.HandleFrontendMessage = func(backendWriter io.Writer, msg pgproto3.FrontendMessage) error {
+		if _, ok := msg.(*pgproto3.Describe); ok {
+			time.Sleep(200 * time.Millisecond)
+		}
+		buf, err := msg.Encode(nil)
+		if err != nil {
+			return err
+		}
+		_, err = backendWriter.Write(buf)
+		return err
+	}
+
+	sql := "select $1::varchar"
+	digest := sha256.Sum256([]byte(sql))
+	psName := "stmt_" + hex.EncodeToString(digest[0:24])
+
+	psd, err := conn.Prepare(ctx, sql, sql)
+	var pgErr *pgconn.PgError
+	require.ErrorAs(t, err, &pgErr)
+	require.Equal(t, "57014", pgErr.Code)
+	require.Nil(t, psd)
+
+	faultyConn.HandleFrontendMessage = nil
+
+	_, err = conn.Exec(ctx, "set statement_timeout = default")
+	require.NoError(t, err)
+
+	var existsOnServer bool
+	err = conn.QueryRow(
+		ctx,
+		"select exists(select 1 from pg_prepared_statements where name = $1)",
+		// Avoid using the prepared statement cache or it will clear the broken statement before we can check for its
+		// existence.
+		pgx.QueryExecModeExec,
+		psName,
+	).Scan(&existsOnServer)
+	require.NoError(t, err)
+	require.True(t, existsOnServer)
+
+	psd, err = conn.Prepare(ctx, sql, sql)
+	require.NoError(t, err)
+	require.NotNil(t, psd)
+}
+
+// https://github.com/jackc/pgx/issues/2640
+func TestPrepareFailedParseSchedulesNoCleanup(t *testing.T) {
+	t.Parallel()
+
+	// A Prepare that fails before Parse completes leaves no statement on the server, so the next Prepare must not
+	// spend a round trip deallocating anything.
+
+	config, err := pgx.ParseConfig(os.Getenv("PGX_TEST_DATABASE"))
+	require.NoError(t, err)
+
+	var faultyConn *faultyconn.Conn
+	config.AfterNetConnect = func(ctx context.Context, config *pgconn.Config, conn net.Conn) (net.Conn, error) {
+		faultyConn = faultyconn.New(conn)
+		return faultyConn, nil
+	}
+
+	ctx := context.Background()
+	conn, err := pgx.ConnectConfig(ctx, config)
+	require.NoError(t, err)
+	defer closeConn(t, conn)
+	require.NotNil(t, faultyConn)
+
+	sql := "select foo"
+	psd, err := conn.Prepare(ctx, sql, sql)
+	require.Error(t, err)
+	require.Nil(t, psd)
+	var pErr *pgconn.PrepareError
+	require.ErrorAs(t, err, &pErr)
+	require.False(t, pErr.ParseComplete)
+
+	var sentClose bool
+	faultyConn.HandleFrontendMessage = func(backendWriter io.Writer, msg pgproto3.FrontendMessage) error {
+		if _, ok := msg.(*pgproto3.Close); ok {
+			sentClose = true
+		}
+		buf, err := msg.Encode(nil)
+		if err != nil {
+			return err
+		}
+		_, err = backendWriter.Write(buf)
+		return err
+	}
+
+	psd, err = conn.Prepare(ctx, "select 1", "select 1")
+	require.NoError(t, err)
+	require.NotNil(t, psd)
+	require.False(t, sentClose)
+}
+
 func TestPrepareBadSQLFailure(t *testing.T) {
 	t.Parallel()