custom listview with only one checkbox is selected one at a time

You need to keep track of selected item and code accordingly.

public class CustomAdapter extends BaseAdapter{
    Integer selected_position = -1;

    @Override
    public View getView(int position, View convertView, ViewGroup parent) {
            // Your Code

            chkbox.setChecked(position==selected_position);

            chkbox.setOnCheckedChangeListener(new OnCheckedChangeListener() {

                @Override
                public void onCheckedChanged(CompoundButton buttonView, boolean isChecked) {
                    if(isChecked)
                    {
                        selected_position =  position;
                    }
                    else{
                         selected_position = -1;
                    }
                    notifyDataSetChanged();
                }
            });
            return convertView;


        }
}

Try to change all item boolean value false exclude selected item after notify adapter and also implement ViewHolder design pattern for ListView performance :

    @Override
    public View getView(final int position, View convertView, ViewGroup parent) {
        ViewHolder holder;
        if(convertView==null){
            holder = new ViewHolder();
            convertView = LayoutInflater.from(context).inflate(R.layout.list,null);
            holder.txt1 = (TextView) convertView.findViewById(R.id.textView1);
            holder.chkbox = (CheckBox) convertView.findViewById(R.id.checkBox1);
            convertView.setTag(holder);
        }else{
            holder = (ViewHolder) convertView.getTag();
        }

        holder.txt1.setText(items.get(position));
        holder.chkbox.setChecked(array[position]);
        holder.chkbox.setOnClickListener(new OnClickListener() {

            @Override
            public void onClick(View v) {
                for (int i=0;i<array.length;i++){
                    if(i==position){
                        array[i]=true;
                    }else{
                        array[i]=false;
                    }
                }
                notifyDataSetChanged();
            }
        });

        return convertView;
    }

    class ViewHolder{
        TextView txt1;
        CheckBox chkbox;
    }